Abstract
The paper describes the numerical simulation of the vertical random vibration of train-slab track-bridge interaction system by means of finite element method and pseudoexcitation method. Each vehicle is modeled as four-wheelset mass-spring-damper system with two-layer suspension systems. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler beams connected with each other by spring and damper elements. The equations of motion for the entire system are derived according to energy principle. By regarding rail irregularity as a series of multipoint, different-phase random excitations, the random load vectors of the equations of motion are obtained by pseudoexcitation method. Taking a nine-span simply supported beam bridge traveled by a train consisting of 8 vehicles as an example, the vertical random vibration responses of the system are investigated. Firstly, the suitable number of discrete frequencies of rail irregularity is obtained by numerical experimentations. Secondly, the reliability and efficiency of pseudoexcitation method are verified through comparison with Monte Carlo method. Thirdly, the random vibration characteristics of train-slab track-bridge interaction system are analyzed by pseudoexcitation method. Finally, applying the 3σ rule for Gaussian stochastic process, the maximum responses of train-slab track-bridge interaction system with respect to various train speeds are studied.
1. Introduction
The dynamic analysis of railway track or bridge subjected to a moving train has long been a subject of great interest in the field of civil engineering [1–10]. As is well known [11–19], rail irregularity, with random nature, is one of the most important factors that can amplify the vibration responses of train-track/bridge interaction system. The dynamic responses in most of the previous researches are usually calculated by using only one time-history sample of rail irregularity. From the point of view of random vibration, those results can merely represent one of a series of possible random samples due to the fact that each time-history sample generated from the PSD function of rail irregularity is random.
With the increasing speed and axle load of train, the random vibration of train-track/bridge interaction system has received more and more attention and is usually analyzed by means of Monte Carlo method (MCM) [12, 16]. In this approach, the time-history samples of dynamic responses are calculated by using a series of samples of rail irregularity firstly, and then the statistical characteristics of responses are obtained according to the maximum values of each sample of dynamic responses. In order to ensure the reliability of the simulation, tens, hundreds, or even thousands of samples must be calculated, which may result in intolerable computer time, especially for the sophisticated train-slab track-bridge interaction system.
Pseudoexcitation method (PEM), originally developed from the field of bridge seismic resistance [20], has been applied to analyze efficiently and accurately the linear time-independent or time-dependent systems with stationary or nonstationary random excitations [21–25]. Lu et al. [23] studied the stationary random vibration responses of vehicle-ballasted track interaction system by PEM in which a time-independent vehicle-track interaction model was adopted. By adopting a train-bridge interaction element with the track structure omitted [26], PEM was applied to research the nonstationary random vibration of time-dependent train-bridge interaction system [24].
With the increasing use of ballastless track on bridge in modern high-speed railway, this paper investigates the vertical random vibration responses of slab ballastless track and bridge subjected to a moving train based on finite element method (FEM) and PEM. The train, slab track, and bridge are regarded as an integrated system. Each moving vehicle is modeled as four-wheelset mass-spring-damper system with two-layer suspension systems possessing 6 degrees of freedom (DOFs) [17]. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler beams [17, 19, 27]. The elasticity and damping properties of the fastener and bridge bearing are represented by discrete springs and dampers. Simultaneously, the elasticity and damping properties of the concrete asphalt layer (CA layer) are represented by continuous springs and dampers [27, 28]. The equations of motion of finite element form for the entire system are derived by means of energy principle [28]. The effects of rail irregularity are regarded as a series of multipoint, different-phase random excitations by taking time lags between the wheels into account. The random excitations between rail and wheels caused by rail irregularity are then transformed into a series of deterministic pseudoharmonic excitation vectors by PEM, so that the random vibration responses of train, slab track, and bridge can be obtained by step-by-step integration method such as Newmark-β method [29] or Wilson- method [30]. Taking a nine-span simply supported beam bridge traveled by a train consisting of 2 motor cars and 6 trailer cars as an example, the suitable number of discrete frequencies of rail irregularity for calculating the random vibration responses of the system is obtained by numerical experimentations, the reliability and efficiency of the random vibration responses obtained by PEM are verified through comparison with those by MCM; the random vibration characteristics of train-slab track-bridge interaction system are analyzed by PEM, and finally the maximum responses of train-slab track-bridge interaction system with respect to various train moving speeds are studied by applying the 3σ rule for Gaussian stochastic process.
2. Models of Train, Slab Track, and Bridge
2.1. Model of Train
Figure 1 shows a train consisting of a series of four-wheelset vehicles moving with constant speed on a slab track structure resting on a series of simply supported beams to model a railway bridge and the two approach subgrades. The train consists of the front and rear motor cars numbered 1 and 2, respectively, and trailer cars numbered from right to left. It is assumed that the downward vertical displacements and clockwise direction rotation of train are taken as positive and that they are measured with reference to their respective static equilibrium positions before moving upon the track concerned.
(a)
(b)
Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. As shown in Figure 1, the carbody rests on the front and rear bogies, each of which in turn is supported by two wheelsets. The carbody is modeled as a rigid body with a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Similarly, each bogie is considered as a rigid body having a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Each wheelset has a mass . The spring and shock absorber in the primary suspension for each wheelset are characterized by spring stiffness and damping coefficient , respectively. Likewise, the secondary suspension between carbody and each bogie is characterized by spring stiffness and damping coefficient . As the carbody is assumed to be rigid, the motions of the th trailer carbody may be described by the vertical displacement and rotation at its center of gravity, where the subscript denotes the trailer car number. Similarly, the motions of the front bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity; the motions of the rear bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity. The motions of the four wheelsets from right to left of the th trailer car may be described by the vertical displacements , , , and , respectively. Therefore, the total number of DOFs for each trailer car is ten. However, it is assumed that each wheelset of all vehicles is always in full contact with the rail in this paper; that is, the vertical displacement of each wheelset is constrained by the displacement of the rail. Consequently, the independent DOFs for each trailer car become six.
Each motor car in the train is also modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. The carbody has a mass and a moment of inertia . Each bogie has a mass and a moment of inertia . Each wheelset has a mass . The primary suspension is characterized by spring stiffness and damping coefficient . Likewise, the secondary suspension is characterized by spring stiffness and damping coefficient . The independent DOFs for each motor car include, too, the carbody vertical displacement and rotation , the front bogie vertical displacement and rotation , and the rear bogie vertical displacement and rotation , where the subscript denotes the motor car number.
2.2. Models of Slab Track and Bridge
As shown in Figure 1, the rail, slab, and bridge girder are all modeled as elastic Bernoulli-Euler beam. The two rails are effectively treated as one in the subsequent analysis. On the basis of FEM, the rail, slab, and bridge girder are all divided into a series of beam elements of equal length . The elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness and dampers with damping coefficient . The elasticity and damping properties of the CA layer underlying the slab are represented by continuous massless springs with stiffness and dampers with damping coefficient . Simultaneously, the elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness and dampers with damping coefficient . It is assumed that the damping of the rail and slab is neglected [19, 31], and the bridge girder has linear viscous damping [15, 32]. In addition, by neglecting axial deformations of the rail, slab, and bridge girder, each node of the rail, slab, and bridge girder has two DOFs, that is, vertical displacement and rotation about an axis normal to the plane of paper. The cubic Hermitian interpolation polynomials are used as shape functions of the rail, slab, and bridge girder elements.
It is assumed also that the downward deflections of rail, slab, and bridge are taken as positive and that they are measured with reference to their respective vertical static equilibrium positions. Let denote the initial vertical irregularities of rail and be measured with reference to smooth profile of rail; that is, , if the top surface of rail is smooth. Likewise, it is considered positive in the downward direction.
3. Equations of Motion for a Train-Slab Track-Bridge Interaction System
By using the energy principle, such as the principle of a stationary value of total potential energy of a dynamic system [28], one can derive the equations of motion written in submatrix for the train-slab track-bridge interaction system as where the subscripts “,” “,” “,” and “” denote the train, rail, slab, and bridge girder, respectively. The displacement vector, the matrices of mass, stiffness, and damping, as well as the load vector for the train, rail, slab, and bridge girder, are explained as follows.
3.1. Displacement Vectors
The total train displacement vector with order can be written as where the superscript “” denotes the transpose of the vector and () and () are the displacement vectors of the th trailer car and the th motor car, respectively, which can be expressed as
The displacement vector of rail with order can be written as where denotes the total number of DOFs of rail.
The displacement vector with order for a series of continuously supported beams to model slab can be written as where () denotes the displacement vector of the th slab, denotes the total number of slab, and denotes the total number of DOFs of all slab. with order and can be expressed as
The displacement vector with order for a series of simply supported beams to model the bridge can be written as where () denotes the displacement vector of the th bridge girder, denotes the total number of bridge girder, and denotes the total number of DOFs of all bridge girder. with order and can be expressed as
3.2. Matrices for Train
The matrices of train are marked with the subscript “.” The mass matrix of train, with order () × (), can be written as where and with order denote the mass matrices of the th trailer car and th motor car, respectively, and can be expressed as
The stiffness matrix of train, with order () × (), can be written as where and with order , denoting the stiffness matrices of the th trailer car and th motor car, respectively, can be expressed as in which denotes half of longitudinal distance between the centers of gravity of trailer car’s front and rear bogies, denotes half of longitudinal distance between the centers of gravity of motor car’s front and rear bogies, denotes half of trailer car’s bogie axle base, and denotes half of motor car’s bogie axle base.
The damping matrix of train with order () × () can be obtained by simply replacing in the corresponding stiffness matrix by .
3.3. Matrices for Rail, Slab, and Bridge
The matrices of rail are marked with the subscript “.” The mass matrix of rail, with order , can be written as with where , with order , represents the overall mass matrix of the rail itself, represents rail mass per unit length, represents the total number of rail elements, represents the local coordinate measured from the left node of a beam element, and , with order , represents the shape function matrix for the th rail element. It should be noted that each element is zero in except for those corresponding to the four DOFs of the two nodes of the th rail element. , with order , represents the overall mass matrix induced by all the wheel masses of trailer cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th trailer car; , , , and , respectively, represent the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the th trailer car and the left node of the rail element upon which the wheelset is acting. It should be noted that each element is zero in except those corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th trailer car is acting. is time dependent as the th wheelset of the th trailer car moves from one position to another within one rail element. As the th wheelset of the th trailer car moves to the next rail element, will shift in position corresponding to the DOFs of the rail element where the th wheelset of the th trailer car is positioned. Similarly, , with order , represents the overall mass matrix induced by all the wheel masses of motor cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th motor car; , , , and , respectively, represent the distances between each wheelset of the th motor car and the left node of the corresponding rail element upon which the wheelset is acting. It should be noted too for that, apart from those elements corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th motor car is acting, all other elements are zero. is also time dependent.
The stiffness matrix of rail, with order , can be similarly expressed in terms of the overall stiffness matrix of the rail itself, overall stiffness matrix induced by all trailer cars, overall stiffness matrix induced by all motor cars, and overall stiffness matrix induced by the stiffness of all fasteners as with where denotes Young’s modulus of the rail and denotes the constant moment of inertia of the rail cross section and the prime differentiation with respect to local coordinate . In the formulation of , the shape function matrix () of order for the rail element is evaluated at the position of the th fastener; denotes the distance between the th fastener and the left node of the rail element containing the th fastener, and denotes the total number of fastener. It should be noted for that, apart from those elements corresponding to four DOFs of the two nodes of the rail element containing the th fastener, all other elements are zero.
With omission of the damping of the rail itself, the damping matrix of rail, with order , can be written in terms of the overall damping matrix induced by all trailer cars, overall damping matrix induced by all motor cars, and overall damping matrix induced by the damping of all fasteners as with
The matrices of slab, with order , are marked with the subscript “” and can be written in terms of the matrices of the slab itself, the matrices induced by fastener on the slab, and the matrices induced by CA layer underlying the slab. The matrices for bridge, with order , are marked with the subscript “” and can be written in terms of the matrices of the bridge girder itself, the matrices induced by CA layer lying on the bridge girder, and the matrices induced by bearing at the support of bridge girder. The matrices of slab and bridge are not derived in detail here but can be obtained by following the similar procedure for derivation of rail matrices.
3.4. Matrices for Train-Rail-Slab-Bridge Interaction
The matrices for train-rail interaction are marked with subscript “” or “.” The stiffness matrices of order , of order , of order , and of order induced by the interaction between the train and rail can be written, respectively, as with in which and represent the stiffness matrices induced by the interaction between the th wheelset of the th trailer car and rail, and and are the corresponding damping matrices. , , , and consist of zero row vectors except for those corresponding to the two DOFs of front bogie of the th trailer car, while , , , and consist of similar column vectors. Accordingly, , , , , , , , and are formed by row or column vectors where the only nonzero elements correspond to the two DOFs of the rear bogie.
The matrices induced by the interaction between motor car and rail can be worked out similarly.
The matrices for rail-slab-bridge interaction can be also derived by following the similar procedure for derivation of train-rail interaction matrices. The matrices for rail-slab interaction, induced by the matrices of fastener between rail and slab, are marked with subscript “” or “.” The matrices for slab-bridge interaction, induced by the matrices of CA layer between slab and bridge girder, are marked with subscript “” or “.”
3.5. Load Vector for Train, Rail, Slab, and Bridge
The load vector of train with order can be written as where the load vector of the th trailer car and and the load vector of the th motor car and with order can be written, respectively, as in which denotes the first derivative of rail irregularity .
The load vector of rail with order can be written as with where denotes the second derivative of rail irregularity and , , , and , respectively, denote the load vectors of the th wheelset of the th trailer car acting upon the rail caused by gravity force of trailer car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity. , , , and are the load vectors of the th wheelset of the th motor car acting upon the rail caused by gravity force of motor car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity, respectively.
In addition, each element for load vector of slab with order and load vector of bridge with order is zero.
Now let
Then, the load vector of the total train-slab track-bridge interaction system can be expressed by the deterministic load vector induced by the train gravity force and the random load vector induced by the rail irregularity can be expressed as
4. Random Vibration Analysis of Train-Slab Track-Bridge Interaction System by PEM
4.1. The Pseudoexcitation for the System
Let be the distance from the left-hand starting point of the track model to the wheel/rail contact point at time ; there exists a transformation from the PSD function of in space domain to the PSD function of in time domain, according to : where , , and denote, respectively, the time frequency (rad/s), the spatial frequency (rad/m), and train speed (m/s). Obviously, .
According to (21)–(26), the pseudoexcitation induced by the rail irregularity can be obtained by PEM: with
4.2. The Calculation of Random Responses of the System
Based on (1) and (26), the equations of motion for a train-slab track-bridge interaction system can be expressed as in which , , and denote mass, damping, and stiffness matrices of the total train-slab track-bridge interaction system, respectively; , , and denote, respectively, displacement, velocity and acceleration vectors of the system.
By assuming the rail irregularity to be a zero mean valued Gaussian random process, the mean value (MV) of arbitrary response of the system is only caused by the train gravity force and can be calculated easily by solving (31) in time domain using a step-by-step integration method. Consider
The arbitrary pseudoresponse of the system caused by can be obtained by solving (32) in frequency domain and time domain. Consider
According to PEM and random vibration theory, the time-dependent power spectral density (PSD) and standard deviation (SD) of arbitrary response of the system can be written as where the superscript “” denotes complex conjugate, denotes the total number of discrete frequencies of rail irregularity, and denotes the th frequency interval of rail irregularity.
4.3. The Solution Procedures
The flowchart for efficiently analyzing the vertical random vibration of train-slab track-bridge interaction system under the action of rail irregularity is shown in Figure 2 by following the above explanation of random vibration analysis by PEM.
5. Case Study
5.1. The Properties of Train-Slab Track-Bridge Interaction System
In order to reduce the boundary effect of the subgrade as far as possible, a nine-span simply supported beam high-speed railway bridge with the span length of 32 m is considered as shown in Figure 1. The central part of railway slab track is supported on bridge, while the left and right parts of the track are supported on subgrades adjacent to the bridge. The track is assumed to be continuous throughout, while the lengths of left and right parts of track considered are, respectively, 260 m and 210 m, in order that the train moving from left to right can not only reach the steady-state response [15] before it runs upon the track on bridge but also move fully out of the right boundary of bridge. The lengths of rail, slab, and bridge girder element are all equal to fastener spacing of 0.65 m. The train comprises front and rear motor cars with the same properties and six identical trailer cars moving with constant velocity . The physical parameters of train, track, and bridge are listed, respectively, in Tables 1 and 2. The Germany high-speed track vertical profile irregularity PSD function [13, 33] is adopted; that is, where rad/m, rad/m, and m·rad.
To solve the equations of motion for the train-slab track-bridge interaction system, the Wilson- method is used with and moving length of 0.1 m of vehicles along track for each time step. In addition, the spatial frequency of the PSD ranges from to rad/m; that is to say, the wavelength of rail irregularity ranges from 1 to 250 m. It should be noted that a series of wavelengths of unequal intervals are selected by the following expression; that is, where denotes the wavelength of rail irregularity () and denotes a constant related to the maximum frequency points .
5.2. Selection of the Suitable Number of Discrete Frequencies of Rail Irregularity
From the aforementioned solution procedures for random analysis of train-slab track-bridge system, one can find that the larger the number of discrete frequencies is selected, the more calculation steps are. In order to save computer time as much as possible and, meanwhile, ensure a high accuracy of solution, a sequence of numerical experimentations is carried out to obtain the suitable number of discrete frequencies of rail irregularity with train running at speed of 69.44 m/s (250 km/h). As increases from 50 to 250 with increment 20, the corresponding is listed in Table 3, based on which the solutions are plotted in Figures 3, 4, 5, 6, 7, and 8. The variation ratio VR1 of the responses of train, track, and bridge and the CPU times for different cases are shown in Table 4. Herein VR1 is defined as (36), in which denote the maximum dynamic response SD of train, track, and bridge with the total number of discrete frequencies being equal to :
According to Yang et al. [15], the dynamic response of the last vehicle of a train should be given much more attention concerning the running safety and riding comfort of the train moving over a bridge because the last vehicle tends to vibrate more violently than the ones ahead. Therefore, “motor car” and “trailer car” considered in the following examples, respectively, mean the rear motor car and the th trailer car connecting with the rear motor car (Figure 1). In addition, “bridge midpoint” and “wheel/rail vertical force,” respectively, are the midpoints of the fifth span for the nine-span bridge and the vertical contact force between the 3rd wheelset of the rear motor car or of the th trailer car and rail.
From Figures 3–8 and Table 4, it is observed that, as the total number of discrete frequencies increases, each of the maximum SD of train, track, and bridge dynamic responses tends to approach a limit value. Comparing the results obtained by 250 discrete frequencies with those by 170 discrete frequencies, the variation ratios VR1 are −0.23% and −0.25% for the vertical acceleration SD of the motor and trailer carbody, respectively, and 1.15%, 1.59%, and −0.19% for the vertical acceleration SD of rail, slab, and bridge girder, respectively. Therefore, one can conclude that the suitable number of discrete frequencies of rail irregularity can be equal to 170.
5.3. Comparing PEM with MCM
The running safety and riding comfort of train have been of great concern in railway engineering for a long time, particularly due to the development of high-speed railway. Several mechanisms that result in the derailment or discomfort of a running train have been identified through analytical and experimental investigations, based on which some indices have been proposed for evaluating the running safety and riding comfort of train [34], such as carbody vertical acceleration, wheel load decrement ratio, Sperling’s ride index, and acceleration of bridge girder. In practice, the maximum dynamic responses are usually of the most interest for us. As is mentioned above, the conventional method of estimation to pick the maximum values from a set of response samples is extremely time consuming. However, the upper and lower boundaries of the responses by PEM can be estimated easily by the MV ± 3 times SD according to the 3σ rule for Gaussian stochastic process [24], since the corresponding MV and SD can be obtained efficiently and accurately according to the aforementioned solution procedures.
For the purpose of comparison, the random responses of the train-slab track-bridge interaction system calculated by PEM, with total discrete frequencies , are compared with those by MCM, with 1000 of rail irregularity samples. Herein, the method proposed by Chen and Zhai [35] is implemented to generate the rail irregularity sample from the PSD function described as (34), and the train is also assumed to pass through the bridge with a constant speed 69.44 m/s. Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 show the estimated upper and lower boundaries of the responses by PEM and the corresponding extreme values of each time-history response obtained by MCM. The variation ratio VR2 of the responses of train, track, and bridge girder, determined according to (37), is shown in Table 5. Consider where denote the extreme values of each time-history response obtained by MCM and denotes the number of rail irregularity samples.
From Figures 9–20 and Table 5, the following observations can be made easily. (1) There exist dramatic variations for the extreme values of each time-history response by MCM, except for that of bridge girder vertical displacement. For example, variation ratio VR2 of vertical acceleration of motor carbody, wheel load decrement ratio of trailer car, Sperling’s ride index of motor car, and vertical acceleration of bridge girder midpoint among the 1000 samples are 280.37%, 120.08%, 39.00%, and 170.05%, respectively. In general, a larger number of rail irregularity samples tend to enlarge the variation ratio VR2. (2) All of extreme values of responses obtained by MCM fall with the corresponding limits of upper and lower boundaries of those by PEM. (3) Compared with MCM, the PEM helps to save computer time drastically. For example, the total CPU times for PEM with 170 discrete frequencies and MCM with just 100 samples are 1.64 hours and 13.12 hours, respectively, on a 2.8 GHz personal computer. The latter equals about eight times the former. That is to say, PEM, in dealing with random vibration of train-slab track-bridge interaction system or evaluating the running safety and riding comfort of train, is not only more reliable but also more efficient than the conventional MCM.
5.4. The Random Vibration Characteristics of Train-Slab Track-Bridge Interaction System
The dominant vibration frequency of train and bridge can be obtained easily according to the PSDs of dynamic responses calculated by PEM. To investigate the random vibration characteristics of train-slab track-bridge interaction system, the solutions obtained by PEM, with 170 discrete frequencies of rail irregularity and three train speeds, are employed.
Figures 21 and 22 exhibit the PSDs of carbody vertical acceleration. As can be seen, the PSDs vary only slightly with time which indicates that the influence of bridge vibration on carbody vibration is insignificant because of the comparatively high mass and flexural rigidity of bridge girder. On the other hand, the PSDs change violently with rail irregularity frequency, implying the great influence of rail irregularity on the dynamic response of carbody. Of interest is that, no matter how the train speed changes, the peak values of PSDs of carbody vertical acceleration appear always approximately, respectively, to be around 0.77~0.86 and 0.69~0.77 Hz for motor car and trailer car, which show good agreement with their own fundamental frequencies of 0.84 and 0.75 Hz.
| (a) m/s (150 km/h) |
| (b) m/s (250 km/h) |
| (c) m/s (350 km/h) |
| (a) m/s (150 km/h) |
| (b) m/s (250 km/h) |
| (c) m/s (350 km/h) |
The similar phenomenon for bogie vertical acceleration can be also observed from Figures 23 and 24. As can be seen, the bogie also vibrates periodically, implying three dominant frequencies of 4.22, 16.79, and 33.49 Hz for m/s, 6.31, 26.99, and 55.82 Hz for m/s and 6.37 (motor car), or 7.92 (trailer car), 39.18, and 78.15 Hz for m/s. It is found that the vertical vibration frequency of bogie, the half of bogie axle base , and the train speeds satisfy the relation , , where relatively large contributions are made by harmonic components with , 2, or 8~15.
| (a) m/s (150 km/h) |
| (b) m/s (250 km/h) |
| (c) m/s (350 km/h) |
| (a) m/s (150 km/h) |
| (b) m/s (250 km/h) |
| (c) m/s (350 km/h) |
The PSDs of vertical acceleration of bridge girder midpoint to the passage of the train are plotted in Figure 25. Similar with the cases of bogie vertical acceleration, there exist in Figure 25 three dominant frequency ranges of 4.7~5.4, 14~18, and 32~37 Hz for m/s, 4.4~6.3, 25~33, and 53~58 Hz for m/s and 4.4~6.8, 35~43, and 72~81 Hz for m/s. Obviously, the first dominant frequency ranges of bridge girder vertical acceleration vary slightly around the fundamental frequency of bridge girder Hz under various train speeds. However, the second and the third dominant frequency ranges increase as the train speed increases, both of whose trends quite coincide with those of bogie vertical acceleration (Figures 23 and 24). Furthermore, the contribution of the latter two ranges of dominant frequency to bridge girder vertical acceleration show an obvious trend of increase for higher train speeds.
| (a) m/s (150 km/h) |
| (b) m/s (250 km/h) |
| (c) m/s (350 km/h) |
From the point of view of structural dynamics, so-called resonance may occur if the wavelengths or frequencies implied by rail irregularity are close to the vehicle or bridge girder frequencies. Therefore, it is especially important to control strictly the rail irregularity with wavelengths of and in high-speed railway track maintenance, where and denote the fundamental frequencies of vertical motion of carbody and bridge girder, respectively. In addition, it is also significant to control the rail irregularity with wavelengths of and to reduce the vibration of bogie and bridge girder as much as possible for high-speed railway.
5.5. Influence of Train Speeds on the Dynamic Responses of Train-Slab Track-Bridge Interaction System
In reality, the train can move over the bridge at various speeds. There exists a need to investigate the dynamic responses of train-track-bridge system under various train speeds, as they may be quite different. The effect of train speeds on the dynamics responses of train-slab track-bridge interaction system is now investigated by PEM with speeds from 0 to 420 km/h at 10 km/h intervals and 170 discrete frequencies of rail irregularity considered. The estimated maximum responses of train, track, and bridge with respect to train speeds are shown in Figures 26, 27, 28, 29, and 30.
The maximum vertical accelerations of carbody under various train speeds are plotted in Figure 26. In general, the maximum response of the train moving over irregular track increases as train speeds increase, with a moderate growth for train speeds over 260 km/h. The reason for existence of such trend is that the exciting frequencies induced by rail irregularity at higher speeds are filtered out by the suspension system of each vehicle in transmission to the carbody. Moreover, both the maximum vertical accelerations of motor and trailer carbody are below the allowable limit of 1.3 m/s2 imposed by China’s provision [33] for speeds up to 420 km/h.
Figure 27 shows Sperling’s ride index computed for the train under various speeds. Similar with the case of carbody vertical acceleration, Sperling’s ride index of both motor car and trailer car increase comparatively slightly for train moving at moderate to high speeds (200–420 km/h), with the former a mite higher than the latter. Compared with the allowable limit values set in China’s provision for different comfort levels [33], Sperling’s ride indices of motor car and trailer car fall with the limit of “tolerate ()” and “good (),” respectively, under the track conditions specified for speeds up to 420 km/h.
Maximum wheel load decrement ratio with respect to train speeds are presented in Figure 28, which shows a trend of increase for higher train speeds. Of interest is the fact that the wheel load decrement ratio of trailer car is always greater than that of motor car. At the meantime, the former increases faster than the later for train speeds over 260 km/h. As a result, the wheel load decrement ratio of trailer car exceeds the limit of allowable 0.65 set in China’s provision for the running safety of train [33] for speeds up to 370 km/h. In contrast, the wheel load decrement ratio of motor car remains to be lower than the allowable limit of 0.65 for speeds up to 420 km/h. From this point of view, the larger axle load is beneficial to ensure the running safety of train.
The impact responses of track and bridge girder traveled by train moving at speeds 0–420 km/h are plotted in Figures 29 and 30, in which the impact factor is defined as , where and , respectively, denote the maximum dynamic and static vertical responses of track or bridge girder at section due to the action of train. As can be seen, the impact factor and vertical acceleration of bridge girder increase generally as the train speed increases, but no phenomenon of train-track-bridge resonance is observed, indicating that the fundamental frequency of bridge girder is suitable under the train properties specified for speeds over 420 km/h. In addition, the impact factor for rail vertical displacement is generally higher than that for bridge girder vertical displacement. The implication is that the extent of amplification for rail vertical displacement is larger than that for bridge girder vertical displacement, because of the comparatively small mass of rail and the effects of elasticity and damping of fastener and CA layer. Of interest is that the bridge girder vertical acceleration is far below the allowable limit of 3.5 m/s2 set in China’s provision under the track conditions specified for speed up to 420 km/h.
6. Summary
In this study, the vertical vibration model of train-slab track-bridge interaction system is firstly established by FEM; the equations of motion for the entire system are then derived by means of energy principle, and simultaneously the excitations between rail and wheels caused by random rail irregularity are transformed into a series of deterministic pseudoharmonic excitation vectors by PEM, and the random vibration responses of train, slab track, and bridge are finally obtained by step-by-step integration method. From the numerical results obtained in this work, the following conclusions can be drawn.(1)In order to save computer time and ensure accuracy of solution as far as possible, some numerical experimentation should be done firstly to obtain the suitable number of discrete frequencies of rail irregularity in the process of random analysis of train-slab track-bridge interaction system.(2)The PEM can be applied not only to study the vertical random vibration characteristics of train-slab track-bridge interaction system but also to evaluate the running safety and riding comfort of train moving over bridge with various speeds, which is more reliable and efficient than the conventional MCM.(3)The PEM may provide a new way to either optimize the design of dynamic parameters of train-slab track-bridge interaction system or provide guidance on the maintenance of track structure according to certain evaluation standard for railway.(4)The equations of motion of vertical interaction system with other types of train, track, and bridge and the corresponding random vibration responses can be also obtained by the method proposed in this paper.(5)In this work, only the vertical random vibration characteristics of train-slab track-bridge interaction system are investigated. In the future work, we plan on further developing the numerical method and extending its range of applicability to model the spatial random vibration characteristics of train-slab track-bridge interaction system.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research summarized in this paper was supported by the Joint Funds of the National Natural Science Foundation of China through Grant nos. U1334203, U1361204, and U1134209, the National Natural Science Foundation of China through Grant no. 51378513, the Program for Changjiang Scholars and Innovative Research Team in University through Grant no. IRT1296, the Science and Technology Foundation of Chinese Railway Corporation through Grants nos. 2012G013-B and 2013G008-E, the Teacher Research Fund of Central South University of China through Grant no. 2013JSJJ021, the China Postdoctoral Science Foundation through Grant no. 2013M530022, and the Major State Basic Research Development Program of China through Grant no. 2013CB036203. These supports are gratefully acknowledged.