A Kalman filter algorithm for identifying track irregularities of railway bridges using vehicle dynamic responses

Mechanical Systems and Signal Processing 138 (2020) 106582

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

A Kalman filter algorithm for identifying track irregularities of railway bridges using vehicle dynamic responses Xiang Xiao a, Zhe Sun a, Wenai Shen b,⇑ a b

School of Transportation, Wuhan University of Technology, Wuhan 430070, China School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 23 October 2019 Received in revised form 13 December 2019 Accepted 16 December 2019

Keywords: Track irregularity identification Railway track monitoring Kalman filter Time-dependent system Vehicle-bridge interaction Unknown input

a b s t r a c t Track irregularities affect the running safety of railway vehicles and ride comfort, hence track irregularity identification using the dynamic responses of in-service vehicles is of great interest. Because the high-speed rail lines mainly consist of bridges in China, vehicle-bridge (VB) interactions which significantly influence the vehicle dynamic responses should be taken into account in the track irregularity identification. This paper proposes a Kalman filter algorithm to identify the track irregularities of railway bridges using vehicle dynamic responses considering the VB interactions in real-time. A state space model is established to represent a time-dependent VB system subjected to unknown track irregularity excitations. A Kalman filter algorithm is proposed to estimate optimally the state vector of the VB system and to identify the track irregularities subsequently. Two numerical examples including a real railway bridge constructed in China are presented to validate the accuracy of the proposed algorithm. A parametric study is also conducted to demonstrate the effects of measurement noise, vehicle running state, parameter uncertainty and model uncertainty on the identification of track irregularities. Comparison results demonstrate that the proposed track irregularity identification algorithm outperforms the conventional approaches mainly because of considering the VB interaction. The proposed algorithm enables efficient monitoring the track irregularities of railway bridges using the acceleration responses of in-service vehicles. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Track irregularities results from both ordinary operation and railway degradation such as ground settlement and sleeper degradation, which significantly affect the ride comfort and even the running safety of railway vehicles. For instance, a high level of track irregularities may cause vehicle derailment and the damage of track-vehicle systems. In urban region, the track irregularities also lead to high-level noise radiation to the surrounding environments. On the other hand, track irregularities are the excitations of vehicles [1–3] and vehicle-bridge (VB) systems [4–6]. Consequently, it is fundamental to identify the track irregularities in both the rail maintenance and the dynamic analysis of VB systems. To date, two methods, i.e. the axle box acceleration method and the inertial reference method, are widely used in commercial track inspection car [7–10]. Both the methods assume that the vertical dynamic displacement of axle boxes represent the track irregularity. The axle box acceleration method estimates the track irregularity by implementing the numerical ⇑ Corresponding author. E-mail addresses: [email protected] (X. Xiao), [email protected] (W. Shen). https://doi.org/10.1016/j.ymssp.2019.106582 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

2

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

double integration of the measured axle box acceleration [10]. However, in the inertial reference method, the track irregularity is regarded as the sum of the car body displacement and the relative displacement between the car body and the axle box. Similar to the axle box acceleration method, the car body displacement is also calculated by conducting the numerical double integration of the measured acceleration. In order to reduce the double integration errors, Lee et al. [11] proposed a mixed filtering approach for identifying the track irregularities based on the measured accelerations of the axle box and bogie. Real et al. [12] proposed an identification method of track irregularities using the inverse Fourier transform technique based on the measured accelerations of axle boxes. However, the numerical error induced by the double integration would be inevitable. Furthermore, the vehicle dynamic responses (including the axle box acceleration /displacement etc.) induced by the VB interaction are pronounced [4–6,13]. Therefore, the identification accuracy of the conventional approaches is most likely to be affected by the VB interactions, when applied to railway bridges. Inverse dynamic analysis is capable of serving as the low-cost and efficient methods for track irregularity monitoring. This is because the track irregularities are essentially the unknown excitations imposing on a vehicle system [14–17]. Czop et al. [14] proposed an approach to monitor the track irregularities using a frequency-domain inverse problem solution based on the measured axle box acceleration. Schenkendorf et al. [16] proposed a novel hybrid approach to monitor the track irregularities using a classical inverse dynamic analysis. However, the above algorithms do not consider the modeling of tracks. As a result, the VB interaction cannot be considered in these approaches. Railway bridges dominate in the high-speed railways in China. It is well known that moving vehicles cause both bridge vibration and vehicle vibration due to the track irregularities. The dynamic interaction between the vehicle and the bridge is strongly coupled, thus the VB interaction is a crucial issue in railway engineering [18–25]. Recently, Zhai et al. [26] conducted a comprehensive review of the VB dynamic interaction and its modeling methods. In the past decades, a great deal of effort has been made to develop the dynamic models for VB systems [27–37]. Almost all of the available vehicle-bridge/ vehicle-track coupling interaction models can be categorized into two types, namely, the elastic wheel-rail contact model and the rigid wheel-rail contact model. For example, Zhai et al. [27], Jin et al. [28], Yang and Hwang [29] and Lee et al. [30] adopt the elastic contact model. On the contrary, Yang et al. [31], Xiao et al. [20,25], Wan and Ni [23], Zeng and Dimitrakopoulos [32], and Lou [33] employ the rigid contact model to treat the contact problem. Both models have merits. In the elastic contact model, the normal wheel-rail contact force is calculated based on the contact stiffness and damping, thus it has high accuracy. However, it may generate numerical problems during the time-integration of the equations of motion thus possibly fail to converge. Hence, exhausted computational labors are needed. Whereas the rigid contact model bypasses the need to estimate the stiffness/damping parameters of contact, as well as the artificial penetration, which indeed results in a much higher computational efficiency [34]. The VB system may resonate under certain conditions [38]. The VB resonance significantly amplifies both vehicle dynamic response and bridge vibration. It is worth noting that the vehicle dynamic response induced by the VB interaction is likely to be larger than that induced by track irregularities. Hence, one should take into account the VB interaction when conducting the track irregularity identification of railway bridges using the vehicle dynamic response. To our best knowledge, however, there are few studies on the track irregularity identification considering the VB interaction. On the other hand, the Kalman filter is a powerful algorithm that has been extensively developed in system identification [39–44]. For example, Lei et al. [39–40] proposed an extended Kalman filter algorithm for identifying the structural physical parameters using limited response output signals. Particularly, many researchers developed the Kalman filter algorithm for force identification [45–48]. Lourens et al. [45] developed an augmented Kalman filter for dynamic force identification, in which the unknown forces included in the state vector can be identified in state vector estimation. Lei et al. [39,46–47] proposed a new algorithm based on Kalman filter to estimate the unknown earthquake ground motions imposed on the tall shear buildings using acceleration measurement signals. Furthermore, a number of Kalman filter algorithms have been proposed for identifying the unknown excitations of nonlinear structural systems [48–51]. To take into account the effect of model parameter uncertainty, Astroza et al. [52–53] conducted a parametric analysis and proposed an improved algorithm. It is well known that track irregularities are the excitations of a vehicle system. Hence the track irregularities can also be treated as unknown input thus being estimated based on the Kalman filter. Recently, Tsunashima et al. [15] proposed a Kalman filter algorithm to identify the track irregularities of the Shinkansen track using the car-body acceleration data based on a vehicle model. As a consequence, the Tsunashima method cannot consider the VB interaction. To date, however, the track irregularity identification of a VB system based on vehicle response has not been addressed yet. Particularly, the mass, damping and stiffness matrices of the VB system considering the VB interaction varies with time, resulting in a time-dependent system. Such complex feature poses a challenge to the track irregularity identification of VB systems. This paper proposes a Kalman filter algorithm to identify the track irregularities of railway bridges based on the dynamic response of vehicles considering the VB interaction. Section 2 establishes a coupled dynamic model of time-dependent VB systems. Section 3 presents the Kalman filter algorithm to estimate optimally the state vector of the VB system and identify the track irregularities. In Section 4, two typical bridges including a real bridge constructed in China are taken as illustrated examples to examine the feasibility and accuracy of the proposed algorithm. The effects of measurement noise and nonstationary running state on the identification results are analyzed in the parametric study. In addition, the parameter uncertainty and model uncertainty are also taken into account and their effects are quantitatively evaluated. The accuracy and robustness of the proposed algorithm are validated in the two numerical examples. The comparison between the proposed

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

3

algorithm and the conventional approaches demonstrates its advantage. The numerical study is followed by a brief discussion. 2. Dynamic modeling of time-dependent VB systems 2.1. Vehicle and bridge model Fig. 1 shows the configuration of a typical time-dependent VB system, which consists of a typical two-span simplysupported bridge and a moving railway vehicle. O-XY represents a global coordinate system (GCS) for modeling the VB system, whose coordinate origin is located at the left end of the bridge. The railway vehicle is simplified as a two-axle massspring-damper system, including a vehicle body, two wheels and the spring-damper connections between them (see Fig. 1(a)). The vehicle body is modeled as a rigid body with two degrees of freedom (DOFs), namely, a vertical displacement yc and a pitching displacement hc. Each wheel is modeled as a rigid body having a vertical displacement ywf for the front wheel and a vertical displacement ywr for the rear wheel. It is assumed that each wheel is always in contact with the rail. As a result, the DOFs of wheels are not independent. Overall, the vehicle model has two independent DOFs, expressed as qv = [yc hc]T. All the parameters of the vehicle model are shown in Table 1. As shown in Fig. 1(b), the vehicle runs with a velocity v(t) and an acceleration a(t) from the left to the right along the longitudinal direction. Although a two-span simply-supported railway bridge is considered in this study, the typical twospan bridge is capable of considering the six possible scenarios representing a train moving on a multi-span continuous railway viaduct. 2.2. Equations of motion for VB systems In this study, the railway bridge is modeled by an Euler beam with equal cross section. By using the modal superposition method [54–56], the equations of motion of the ith (i = 1–2) bridge span can be written as

€ bi þ C bbi q_ bi þ K bbi qbi ¼ f bi M bbi q

ð1Þ

where Mbbi, Cbbi and Kbbi denote the mass, damping and stiffness matrices of the ith bridge span, respectively; qbi denotes the generalized DOF vector in modal coordinate, fbi denotes the dynamic loads acting on the ith bridge-span due to the moving vehicle. The terms of Eq. (1) can be expressed as follows


T qbi ¼ q1;i ðt Þ q2;i ðt Þ    qm;i ðt Þ

ð2Þ

θc

Fig. 1. Configuration of a typical time-dependent VB system.

4

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Table 1 Parameters of the vehicle model and the corresponding values in numerical example 1 and 2. Variable

Description

Unit

Value

mc Jc mwf mwr L1 L2 Lc k1 k2 c1 c2

Mass of vehicle body Moment of inertia of vehicle body Mass of front wheel Mass of rear wheel Distance of front axle from vehicle mass center Distance of rear axle from vehicle mass center Distance between front and rear axles Stiffness of front spring Stiffness of rear spring Damping coefficient of front damper Damping coefficient of rear damper

kg kgm2 kg kg m m m N/m N/m Ns/m Ns/m

4.18  2.08  1.78  1.78  8.00 8.00 16.00 1.18  1.18  3.92  3.92 

104 106 103 103

106 106 104 104

  8 mb L mb L >    m2b L > < M bbi ¼ diag 2 2 C bbi ¼ mb L  diag ð f1 x1 f2 x2    fm xm Þ > >   : K bbi ¼ m2b L  diag x21 x22    x2m

ð3Þ

8 0 < xf 6 L c f ¼ N T1;x¼xf pf ðtÞ; f b2 ¼ 0 > > > b1 > > T T > > f b1 ¼ N 1;x¼xf pf ðtÞ þ N 1;x¼xr pr ðt Þ; f b2 ¼ 0 Lc < xf 6 L > > > > T < f ¼ NT L < xf 6 L þ Lc b1 1;x¼xr pr ðt Þ; f b2 ¼ N 2;x¼xf pf ðt Þ > T T > f b1 ¼ 0; f b2 ¼ N 2;x¼xf pf ðtÞ þ N 2;x¼xr pr ðtÞ L þ Lc < xf 6 2L > > > > > T > > 2L < xf 6 2L þ Lc f ¼ 0; f b2 ¼ N 2;x¼xr pr ðt Þ > > : b1 f b1 ¼ 0; f b2 ¼ 0 xf ¼ 0 or xf > 2L þ Lc

ð4Þ

The definitions of the variables refer to Appendix C in this study unless otherwise stated. The shape function vector Ni of the ith bridge span is given by


 N i ¼ /1;i ðxÞ /2;i ðxÞ    /m;i ðxÞ

ð5Þ

The nth mode shape function and frequency of the ith bridge span are conveniently expressed in the form

/n;i ðxÞ ¼ sinðnpðx=L  iÞÞ; xn ¼ ðnp=LÞ2

pffiffiffiffiffiffiffiffiffiffiffiffiffi EI=mb

ð6Þ

The vertical displacement yi of the ith bridge span can be expressed as

yi ðx; tÞ ¼

m X

qn;i ðt Þ/n;i ðxÞ ¼ N i qbi

ð7Þ

n¼1

According to the assumption that each wheel keeps always in contact with the rail, the vertical displacements ywf and ywr of the two wheels can be expressed as



ywf ¼ N i;x¼xf qb þ Rf ywr ¼ N i;x¼xr qb þ Rr

ð8Þ

where Rf and Rr denote the track irregularities at the front wheel and the rear wheel, respectively. The dynamic loads pf and pr consist of the static axle loads, the inertia force of wheels, the damping force of dampers and the elastic force of springs, which are given by

8     > €wf þ c1 y_ c þ h_ c L1  y_ wf þ k1 yc þ hc L1  ywf < pf ðt Þ ¼ wf  mwf y   > : pr ðt Þ ¼ wr  mwr y €wr þ c2 y_ c  h_ c L2  y_ wr þ k2 ðyc  hc L2  ywr Þ

ð9Þ

where wf and wr denote the static loads of the front and rear axles, respectively; mwf and mwr denote the mass of the front and rear wheels, respectively; L1 and L2 denote the distances from the mass center of the vehicle body to the front and rear axles, respectively; k1 and c1 denote respectively the stiffness and damping coefficient of the spring-damper connection at the front axle, and k2 and c2 denote the counterparts at the rear axle. According to D’Alembert’s principle, the equations of motion of the vehicle are expressed as

5

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

8       > €c þ c1 y_ c þ h_ c L1  y_ wf þ c2 y_ c  h_ c L2  y_ wr þ k1 yc þ hc L1  ywf þ k2 ðyc  hc L2  ywr Þ ¼ 0 < mc y       > : J c €hc þ c1 L1 y_ c þ h_ c L1  y_ wf  c2 L2 y_ c  h_ c L2  y_ wr þ k1 L1 yc þ hc L1  ywf  k2 L2 ðyc  hc L2  ywr Þ ¼ 0

ð10Þ

where mc and Jc denote the mass and moment of inertia of the vehicle body, respectively. By using Eqs. (1), (7)–10), the equations of motion of the VB system can be obtained by eliminating the variables pf and pr, i.e.

9 2 38 €v > M vv 0 0 C v b1 C vv > >q > > < = 6 7> 6 7 € 6 6   M bb1 þ M bb1 0 7 qb1 þ 6 C b1v C bb1 þ C bb1 6 0 > 5> 4 4 > > > > :€ ; 0 0 M bb2 þ M bb2 C b2v 0 qb2 8 8 9 9 2 3 K v b1 K v b2 K vv > > > qv > > pv > > > > > < < = > = 6 7> 6  7  0 þ 6 K b1v K bb1 þ K bb1 7 qb1 ¼ pb1 > > > 4 5> > > > > > > : :  > ; > ; pb2 K v b2 0 K bb2 þ K bb2 qb2 2

9 38 q_ > > > > v > > = 7< 7 _ 0 7 qb1 > 5> > > > :_ > ; C bb2 þ C bb2 qb2 C v b2

ð11Þ

in which

M vv ¼ diag ðmc ; Jc Þ; C vv ¼
pv ¼ k1 Rf þ k2 Rr

c 1 þ c2

c 1 L1  c 2 L2

c1 L1  c2 L2

cL21 þ cL22

k 1 L 1 Rf  k 2 L 2 Rr



; K vv ¼

k1 þ k2

k1 L1  k2 L2

k1 L1  k2 L2

kL21 þ kL22

T

 ð12Þ ð13Þ

where Mvv, Cvv and Kvv denote the mass, damping and stiffness matrices of the vehicle, respectively; pv and p*bi(i = 1–2) denote the load vectors of the vehicle and the bridge, respectively. The matrices or vectors M*bbi, C*bbi, K*bbi, C*vbi, C*biv, K*vbi, K*biv and p*bi(i =1–2) can be expressed according to the following six scenarios, as shown in Fig. 1(b). Scenario I: When the moving distance xf of the vehicle satisfies 0 < xf  Lc

8  T > > > M bb1 ¼ mwf N 1;x¼xf N 1;x¼xf > > > > >  > > C bb1 ¼ c1 N T1;x¼xf N 1;x¼xf > > > > > > > > K bb1 ¼ k1 N T1;x¼x N 1;x¼xf > > f > > > > h iT <  T K v b1 ¼ K b1v ¼  k1 N T1;x¼xf L1 N T1;x¼xf > > > > h iT >   T > >  T T > > C v b1 ¼ C b1v ¼  c1 N 1;x¼xf L1 N 1;x¼xf > > > > > > T T >  > > > pb1 ¼  k1 Rf N 1;x¼xf þ wf N 1;x¼xf > > > > > : M  ¼ C  ¼ K  ¼ 0; K  ¼ K  T ¼ C  ¼ C  T ¼ 0; p ¼ 0 bb2 bb2 b2v v b2 v b2 bb2 b2v b2

ð14Þ

Scenario II: When the moving distance xf of the vehicle satisfies Lc < xf  L

8 > M bb1 ¼ mwf N T1;x¼xf N 1;x¼xf þ mwr N T1;x¼xr N 1;x¼xr > > > > > > > > > C bb1 ¼ c1 N T1;x¼xf N 1;x¼xf þ c2 N T1;x¼xr N 1;x¼xr > > > > > > > > K bb1 ¼ k1 N T1;x¼xf N 1;x¼xf þ k2 N T1;x¼xr N 1;x¼xr > > > > > > h iT h iT <  T K v b1 ¼ K b1v ¼ k1 N T1;x¼xf L1 N T1;x¼xf  k2 N T1;x¼xr L2 N T1;x¼xr > > > > h iT h iT >   > > > C v b1 ¼ C b1v T ¼ c1 N T1;x¼x L1 N T1;x¼x  c2 N T1;x¼x L2 N T1;x¼x > > r r f f > > > > > T T T T >  > > > pb1 ¼  k1 Rf N 1;x¼xf  k2 Rr N 1;x¼xr þ wf N 1;x¼xf þ wr N 1;x¼xr > > > > > : M  ¼ C  ¼ K  ¼ 0; K  ¼ K  T ¼ C  ¼ C  T ¼ 0; p ¼ 0 v b2 v b2 bb2 bb2 b2v bb2 b2v b2

Scenario III: When the moving distance xf of the vehicle satisfies L < xf  L + Lc

ð15Þ

6

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

8  8  M bb2 ¼ mwf N T2;x¼xf N 2;x¼xf > M bb1 ¼ mwr N T1;x¼xr N 1;x¼xr > > > > > > > > > >  T > > C bb2 ¼ c1 N T2;x¼xf N 2;x¼xf > > > C bb1 ¼ c2 N 1;x¼xr N 1;x¼xr > > > > > > >  T  T > > > > < K bb1 ¼ k2 N 1;x¼xr N 1;x¼xr < K bb2 ¼ k1 N 2;x¼xf N 2;x¼xf h i h iT T  T   T ;  T T > > K v b1 ¼ K b1v ¼  k2 N T1;x¼xr L2 N T1;x¼xr K ¼ K ¼  k N L N > > 1 1 v b2 v b2 2;x¼x 2;x¼x > > f f > > > > h iT h iT > > > >   T T > C  ¼ C  T ¼ c N T >  T T > > L N 2 C v b2 ¼ C b2v ¼  c1 N 2;x¼xf L1 N 2;x¼xf > > b1v v b1 1;x¼xr 2 1;x¼xr > > > > > > : > T T  : pb1 ¼  k2 Rr N 1;x¼xr þ wr N 1;x¼xr pb2 ¼  k1 Rf N T2;x¼x þ wf N T2;x¼x f

ð16Þ

f

Scenario IV: When the moving distance xf of the vehicle satisfies L + Lc < xf  2L

8  M bb2 ¼ mwf N T2;x¼xf N 2;x¼xf þ mwr N T2;x¼xr N 2;x¼xr > > > > > > > C bb2 ¼ c1 N T2;x¼xf N 2;x¼xf þ c2 N T2;x¼xr N 2;x¼xr > > > > > K  ¼ k NT T > > 1 2;x¼xf N 2;x¼xf þ k2 N 2;x¼xr N 2;x¼xr bb2 > > > h iT h iT <  T K v b2 ¼ K b2v ¼  k1 N T2;x¼xf L1 N T2;x¼xf  k2 N T2;x¼xr L2 N T2;x¼xr > > h iT h iT >  T > > > C v b2 ¼ C b2v ¼  c1 N T2;x¼xf L1 N T2;x¼xf  c2 N T2;x¼xr L2 N T2;x¼xr > > > > > T T T > p ¼  k R N T > > 1 f 2;x¼xf  k2 Rr N 2;x¼xr þ wf N 2;x¼xf þ wr N 2;x¼xr b2 > > >    T : T M bb1 ¼ C bb1 ¼ K bb1 ¼ 0; K v b1 ¼ K v b1 ¼ C b1v ¼ C b1v ¼ 0; pb1 ¼ 0

ð17Þ

Scenario V: When the moving distance xf of the vehicle satisfies 2L < xf  2L + Lc

8 > M bb2 ¼ mwr N T2;x¼xr N 2;x¼xr > > > > > > > > C bb2 ¼ c2 N T2;x¼xr N 2;x¼xr > > > > > > >  T > > K bb2 ¼ k2 N 2;x¼xr N 2;x¼xr > > > > > h iT <   T K v b2 ¼ K b2v ¼  k2 N T2;x¼xr L2 N T2;x¼xr > > > > h iT >  T > > > C v b2 ¼ C b2v ¼  c2 N T2;x¼xr L2 N T2;x¼xr > > > > > > > T T  > > > pb2 ¼  k2 Rr N 2;x¼xr þ wr N 2;x¼xr > > > > > : M  ¼ C  ¼ K  ¼ 0; K  ¼ K  T ¼ C  ¼ C  T ¼ 0; p ¼ 0 bb1 bb1 b1v bb1 b1v v b1 v b1 b1

ð18Þ

Scenario VI: When the moving distance xf of the vehicle satisfies xf = 0 or xf > 2L + Lc

 T  T M bbi ¼ C bbi ¼ K bbi ¼ 0; K v bi ¼ K biv ¼ C v bi ¼ C biv ¼ 0; pbi ¼ 0

ð19Þ

The matrices or vectors M*bbi, C*bbi, K*bbi, C*vbi, C*biv, K*vbi, K*biv and p*bi represents the interactions between the vehicle and the bridge, which are essentially time-dependent when the vehicle is moving on the bridge. 3. Identification algorithm of track irregularities This section establishes the discrete-time state-space model of a VB system for inverse dynamic analysis. The corresponding characteristics are analyzed in detail to explore the difference from traditional structures. Subsequently, we propose a Kalman filter algorithm to identify the track irregularities of railway bridges in real time. 3.1. State-space model of VB systems The equations of motion of a VB system (Eq. (11)) can be rewritten as

€ þ C ðt Þq_ þ K ðtÞq ¼ pðt Þ M ðt Þq

ð20Þ

where q denotes the generalized DOF vector of the VB system; M, C and K denote the mass, damping and stiffness matrices, respectively; p denotes the load vector. According to Eqs. (13)–(19), the load vector p of the VB system can be rewritten as

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

7

pðtÞ ¼ f ðt Þ þ F ðtÞ  R

ð21Þ

8 9 2 3 0 > k2 k1 > > > > < 0 > = 6k L  k L 7 Rf 2 27 6 1 1 f ¼ ;F ¼ 6 7; R ¼ > > 4 5 g Rr n > b1 b1 > > > : ; gb2 nb2

ð22Þ

where f is the static axle’s load vector; R is the track irregularity vector; F is defined as a input matrix of track irregularities;

gbi denotes the static axle’s load subvector acting on the ith span; nbi represents the input submatrix of the track irregularities on the ith span. The expressions of gbi and nbi are given in Appendix A. Eq. (21) indicates that the load vector p of the VB system consists of the static axle’s load vector f and the load vector FR q of the track irregularities. Defining a state vector x ¼ _ , the equations of motion (Eq. (20)) of the VB system can be conq verted into the following continuous-time state equations

x_ ðt Þ ¼ Aðt Þxðt Þ þ Bðt Þf ðtÞ þ EðtÞRðt Þ

ð23Þ

in which

Aðt Þ ¼

0 M 1 K

 

 0 0 ; E ð t Þ ¼ ; B ð t Þ ¼ M 1 F M 1 C M 1 I

ð24Þ

It should be noted that the matrices A, B and E and vectors f and R are all time-dependent. Considering the time discretization of a measured signal, it is necessary to transform Eq. (23) into discrete-time equations [45,57]. Assuming that the matrices A, B and E and vectors f and R are all constant in a time step, the general solution of Eq. (23) in an arbitrary time step [(k-1)Dt, kDt] can be expressed as

Z

t

xðtÞ ¼ expðA  ðt  t0 ÞÞ  xðt 0 Þ þ t0

ðexpðA  ðt  sÞÞðBf þ ERÞÞds

ð25Þ

Thus the discrete-time state equations with considering system noise are given by

xk ¼ Uk1 xk1 þ U k1 f k1 þ Xk1 Rk1 þ wk

ð26Þ

1 Uk1 ¼ eAk1 Dt ; U k1 ¼ ðUk1 - I ÞA1 k - 1 Bk1 ; Xk1 ¼ ðUk1 - I ÞAk - 1 E k1

ð27Þ

where the matrices and vectors with subscript k or k-1 denote the corresponding values at time t = kDt or (k-1)Dt, respec  tively. The system noise vector wk is formally a white noise process with the covariance E wk wTj ¼ Q dkj , where dkj is the Kronecker delta. The displacement responses of the vehicle DOFs are inconvenient to be measured. On the contrary, the relative displacement between the two different rigid bodies is easily measured by displacement sensors. In addition, it should be noted that the measured responses related to the DOFs of wheels are constrained by the bridge (see Eq. (7)). Thus, the continuous-time observed equations of on-board measurement can be expressed by the generalized DOF vector q of the whole VB system, i.e.

€ þ C v ðt Þq_ þ C d ðt Þq þ C r ðtÞR y ¼ C a ðt Þq

ð28Þ

where y is the observed vector; Ca(t), Cv(t), Cd(t) and Cr(t) are the acceleration, velocity, displacement and track irregularity output matrices, respectively. € , Eq. (28) can be rewritten as By using Eq. (20) to eliminate the term with q

y ¼ H ðt Þx þ Gðt Þf þ Dðt ÞR
8 1 > < H ðt Þ ¼ C d  C a M K 1 DðtÞ ¼ C a M F þ C r > : GðtÞ ¼ C a M 1

ð29Þ C v  C a M 1 C

 ð30Þ

Notably, the matrices H(t), G(t) and D(t) are time-dependent. In real-word applications, the noise level may be significant. Hence, Eq. (29) can be transformed into the discrete-time observed equations with measurement noise, i.e.

yk ¼ H k xk þ Gk f k þ Dk Rk þ v k

ð31Þ

where vk is a measurement noise vector, which is assumed as zero-mean Gaussian white noise with the covariance   E v k v Tj ¼ Pdkj .

8

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

€c and pitching acceleration € In this study, total two dynamic responses of the vehicle, including the vertical acceleration y h of the vehicle body, are measured for track irregularity identification, and the observed vector y can be expressed as
 €c T . As a result, the martrices Hk, Dk and Gk in the observed equation Eq. (31) can be given and listed in Appendix € h y¼ y c

B. Based on the above analysis, the characteristics of the state-space model of VB systems are summarized as follows: 1) The coefficient matrices (Uk, Uk, Xk, Hk, Gk and Dk.) in the discrete-time state equation and observed equation are time-dependent. 2) The input of VB systems consists of two parts, namely the static axle’s loads of the moving vehicle and the track irregularity excitations. The former is known and the latter is to be identified. 3) The observed vector of VB systems contains only the partial output (partial vehicle response). 4) Both the state equation and observation equation contain the unknown input (the track irregularity excitation). Therefore, the track irregularity identification is essentially an inverse dynamic analysis of a time-dependent system with partially unknown excitations using partial output measurements (see Fig. 2). It is intrinsically different from the inverse dynamic analysis of traditional time-independent systems. 3.2. A Kalman filter algorithm for the estimation of track irregularities The Kalman filter algorithm, also known as linear quadratic estimation, is widely used in numerous applications [58,59], such as tire-road friction coefficient identification [58], system identification and damage detection [43,49,51–53,59], and force identification [44,45–50]. More recently, Strano and Terzo [60–62] proposed a real-time estimation method of the wheel-rail contact force based on the extended Kalman filter. However, to the best of our knowledge, there are few studies on using the Kalman filter algorithm to identify the track irregularities of a time-dependent VB system. This section proposes a new Kalman filter algorithm to conduct the inverse dynamic analysis of the time-dependent VB system. According to Eq. (31), the track irregularity vector can be estimated by

Rk ¼ Dþk ðyk  H k xk  Gk f k  v k Þ

ð32Þ T

where the matrix Dk is assumed to satisfy that its product (D kDk) is nonsingular; and the matrix  1 T DTk . sponding generalized inverse matrix, i.e. Dþ k ¼ Dk Dk

D+k

denotes the corre-

Substitute Eq. (32) into Eqs. (26) and (31), we have 



xk ¼ Uk1 xk1 þ U k1  Xk1 Dþk1 v k1 þ wk

ð33Þ

      yk ¼ I  Dk Dþk H k xk þ I  Dk Dþk Gk f k þ Dk Dþk yk þ I  Dk Dþk v k

ð34Þ

in which 

Uk1 ¼ Uk1  Xk1 Dþk1 H k1 ;

ð35Þ

   U k1 ¼ U k1  Xk1 Dþk1 Gk1 f k1 þ Xk1 Dþk1 yk1

ð36Þ 

^k1 of the state vector at the last time (t = (k  1)Dt) is known, the predictions xk Assuming that the optimal estimation x 

and yk of the state vector and observed vector at the current time (t = kDt) are given by 





^k1 þ U k1 xk ¼ Uk1 x

ð37Þ

    yk ¼ ðI  Dk Dþk ÞH k xk þ I  Dk Dþk Gk f k þ Dk Dþk yk

ð38Þ

^k of the state vector can be determined by the measurement vector yk, i.e. The optimal estimation x

Fig. 2. Inverse dynamic analysis for time-dependent VB systems.

9

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

        ^k ¼ xk þ Lk yk  yk ¼ xk þ Lk Wk yk  H k xk  Gk f k x

ð39Þ

where the symmetric matrix Wk satisfies Wk ¼ I  Dk Dþ k ; Lk is a coefficient matrix for the optimal estimation. It is noted that the matrix Wk in Eq. (39) should be a nonzero matrix to achieve the optimal estimation of the state vector. As a result, the number of the measured variables in the observed vector should be bigger than the number of unknown inputs. According to Eqs. (33) and (39), the optimal estimated error vector

   ^k ¼ ðI  Lk Wk Hk Þ xk  xk  Lk Wk v k ^ek ¼ xk  x

It is assumed that the noise vector vk and error vector ^ k can be expressed as covariance matrix P

^

ek is given by ð40Þ

^

ek ¼ xk  x^k are independent of each other, the optimal estimated


 ^ k ¼ E ^ek ^eT ¼ ðI  Lk Wk H k ÞP k ðI  Lk Wk H k ÞT  Lk Wk PWT LT P k k k

ð41Þ



where P k denotes the predicted covariance matrix. Notably, the optimal coefficient matrix Lk is capable of minimizing the total error covariances of the all state variables [63–64], which can be expressed by the trace function of the covariance matrix, i.e.

  h i    ^ k ¼ tr ðI  Lk Wk H k ÞP k ðI  Lk Wk H k ÞT  tr Lk Wk PWT LT tr P k k

ð42Þ

^ k ) with respect to Lk and setting the result equal to zero matrix, we have Taking the derivative of the trace function tr(P

_  @tr P k @Lk

¼

    h i 2 Lk Wk H k P k H Tk  P  P k H Tk WTk ¼ 0

ð43Þ

The sufficient and non-essential condition for satisfying Eq. (43) is given by

    Lk Wk H k P k H Tk  P  P k H Tk ¼ 0

ð44Þ

   For simplicity, a gain matrix Lk ¼ Lk Wk instead of the matrix Lk is herein defined. Assuming the matrix H k P k H Tk  P is nonsingular, the optimal gain matrix L*k can be expressed as 

1



Lk ¼ P k H Tk ðH k P k H Tk  PÞ

ð45Þ

^ k can be simplified as Thus, the optimal estimated covariance matrix P 

^ k ¼ ðI  L H k Þ P P k k

ð46Þ ^

It is assumed that the noise vector wk is independent of the noise vector vk and the error vector ek , the recursive equation 

with respect to the predicted covariance matrix P k can be determined by using Eqs. (33) and (37), i.e.

  T      ^k ¼ Uk1 P P k ¼ E xk  xk xk  xk

 T - 1 Uk1

þ Ck1 PCTk1 þ Q

ð47Þ

where the coefficient matrix Ck1 satisfies Ck1 ¼ Xk1 Dþ k1 . ^ 0 given in prior establish a new Kalman filter algorithm for esti^0 and P Eqs. (37), (39), (45)–(47) with the initial vectors x mating optimally the state vector of the VB system in real time. To identify the track irregularities, Eq. (31) can be rewritten as

^k þ Gk f k þ Dk Rk þ H k ^ek þ v k yk ¼ H k x

ð48Þ

By using the least square method, the optimal estimation of track irregularities is given by

^ k ¼ ðDT W k Dk Þ1 Dk W k ðy  H k x ^k  Gk f k Þ R k k

ð49Þ

where Wk is a weight matrix. The optimal estimation of Wk is given by [64]

 1  1 T ^ k HT þ P W k ¼ E½ðH k ^ek þ v k ÞðH k ^ek þ v k Þ  ¼ Hk P k where P denotes the noise covariance matrix.

ð50Þ

10

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

calculate calculate

Fig. 3. Procedure for estimating track irregularities.

Based on the above theoretical derivation, a general procedure (see Fig. 3) for identifying the track irregularities can be summarized, i.e. ^ 0 . In this ^0 and covariance matrix P 1) Set the initial conditions. For time step k =1, set the values of the initial state vector x ^ ^ study, the values of the initial state vector and covariance matrix are respectively set as x0 ¼ 0 and P 0 = I. 2) Establish the property matrices M, C and K and load vector p of the VB system for the current time (see Eqs. (11)–(19) and (22)). 3) Calculate the coefficient matrices Uk, Uk, Xk, Hk, Gk and Dk and input vector fk of discrete- time state-space equations and observed equations for the current time step (see Eqs. (24), (27) and (30)).

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

11



4) Determine the predicted state vector xk by the optimal estimation at the last time step using Eq. (37); determine the 

predicted covariance matrix P k by the optimal estimation at the last time step using Eq. (47). ^k by 5) Establish the gain matrix L*k at the current time step using Eq. (45); calculate the optimal estimated state vector x the measured vector yk using Eq. (39). ^ k at the current time step using Eq. (46). 6) Calculate the optimal estimated covariance matrix P ^ k by the state vector x ^k 7) Calculate the weight matrix Wk, then determine the optimal estimated track irregularities R using Eq. (49).

8) If the total step number k is less than the preset number Ns, let k =k+1 and return to step 2 for the next time step. Otherwise, stop the procedure. 4. Numerical examples 4.1. Example 1: Validation of the proposed method A two-span simply-supported railway bridge as shown in Fig. 1(b) is taken as the example to verify the validity of the proposed method. Each span of the bridge is L=25 m; the mass per unit length is mb = 2.303  103kg/m; the elastic modulus and the sectional moment of inertia are E = 2.87  109N/m2 and I = 2.9 m4, respectively. The vehicle parameters are shown in Table 1. The vehicle moves from the left end to the right end with a constant velocity of v = 100 km/h. In this numerical € of the vehicle body, the pitching acceleration € h of the vehicle example, total two variables, i.e., the vertical acceleration y c

body are measured for identifying the track irregularities. The dynamic responses of the VB system were computed by the forward dynamic analysis using a given track irregularities. In this example, the track irregularities are assumed to satisfy the sinusoidal function, i.e. R(x) = Asin(2px/k). The amplitude A and wavelength k are 4 mm and 10 m, respectively. In addition, the measurement noise is modelled as a white-noise process with zero mean value. In this example, the noise level, defined as the ratio of the root mean square of measurement noises to that of signals, is set to 5%, and the covariance matrix of noises is determined as P = diag(3.29 m2/s4 0.43 rad2/ s4)  10-5. Fig. 4 shows the time histories of the observation vector y. In theory, the calculation accuracy increases with increasing the number m of the bridge vibration modes used. The effect of the parameter m on the identification results is herein investigated. Total three cases are considered in this numerical example, i.e., m = 1, m = 3, and m = 5, which indicates that the first vibration mode, the first three vibration modes, and the first five vibration modes of the bridge were used in analysis, respectively. Figs. 5–7 show the estimated dynamic responses of the VB system and the identified track irregularities in the three cases. From Fig. 5(a), (b) and 6, it can be found that the differences between the estimated responses and the exact values decrease rapidly with increasing m. Particularly, the estimations agree well with the exact values when m = 5. Similarly, the proposed Kalman filter algorithm accurately identifies the track irregularities when the first five vibration modes of the bridge are used in the computation, as shown in Fig. 7. As shown in Figs. 5(b) and 7, there are unstable oscillations at the beginnings of both the estimated bridge responses and ^ 0 . Importantly, the ^0 and matrix P identified track irregularities, which are mainly induced by the initial values of the vector x oscillations disappear quickly and each curve tends to match the exact values. This phenomenon indicates that the initial conditions have little influence on the identification results.

Fig. 4. Time histories of observation vector (measurement noise level: 5%).

12

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Fig. 5. Comparison of the exact and estimated responses.

Fig. 6. Comparison of the exact and estimated acceleration responses at bridge midspan.

Fig. 7. Comparison of the exact and identified track irregularities (with different vibration modes) (The blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In order to verify the validity of the proposed method under various wavelengths of track irregularity and vehicle velocities, four representative velocities and wavelengths are considered in this study, as shown in Table 2. The wavelength k varies from 10 m to 40 m with a step of 10 m, covering the wavelength band of track irregularities in real railway. And the vehicle velocity v varies from 50 km/h to 250 km/h with a step of 50 km/h, locating in the velocity range in real railway.

13

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582 Table 2 Identification errors at the peak location of track irregularities (example 1). Wavelength k

Identification error (%) v = 50 km/h

v = 100 km/h

v = 150 km/h

v = 200 km/h

v = 250 km/h

10 20 30 40

3.2 4.0 2.8 3.9

2.4 2.5 2.2 2.3

2.3 2.5 2.4 2.8

2.1 2.5 1.8 2.5

2.2 2.4 2.0 2.6

m m m m

To quantify the error of track irregularity identification, two simple indexes are defined in this study. The first index is the      identification error at the peak location of track irregularity curves, i.e. ðRðxp Þ  Rðxp ÞÞ= Rðxp Þ  100%, where Rðxp Þ, Rðxp Þ and xp represent the identified result, the exact value and the peak location of track irregularity curve, respectively. The sec     ond index is the maximum absolute identification error over the entire X-coordinate range, i.e. max RðxÞ  RðxÞ , of which the unit is mm. Table 2 shows the identification errors at the peak location of the exact track irregularity curve. The identification errors in all cases are all smaller than 5%, which validates the high accuracy of the proposed algorithm. 4.2. Example 2: Application to a real railway bridge A two-span simply-supported prestressed concrete railway bridge constructed in China is taken as the example to investigate applicability of the proposed method, as shown in Fig. 1(b). A single span is L = 32 m; the mass per unit length is mb = 9. 4  103 kg/m; the elastic modulus and the moment of inertia are E = 3.45  1010 N/m2 and I = 3.2 m4, respectively; the damping ratio f of each bridge mode is set to 0.02. Fig. 8 shows the bridge girder with a box section. The example bridge with the span of 32 m has been widely used in real railway in China, such as Hangzhou-Changsha high-speed railway. The vehicle parameters and observed variables are identical to those of the example 1. The first eight vibration modes of the bridge are used in the analysis, and the corresponding frequencies are shown in Table 3. Fig. 9 shows the actual track irregularities used in this numerical example, which were measured from a practical railway of China. 4.2.1. Effect of measurement noise In this case, the vehicle moves from the left end to the right end of the bridge with a constant velocity of v = 250 km/h. To investigate the effect of measurement noise on the identified results for the real railway bridge, three measurement noise levels, i.e. 1%, 5% and 10% are taken into account. The corresponding covariance matrices of noises are determined as P = diag(2.06 m2/s4 0.08 rad2/s4)  107, P = diag(5.14 m2/s4 0.19 rad2/s4)  106 and P = diag (2.06 m2/s4 0.08 rad2/ s4)  105, respectively.

Fig. 8. Section of bridge girder (Unit:mm).

Table 3 Natural frequency of the bridge (example 2). Order of mode

Frequency (Hz)

Order of mode

Frequency (Hz)

1 2 3 4

5.26 21.02 47.32 84.11

5 6 7 8

131.43 189.24 257.67 336.45

14

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Fig. 9. Track irregularity in example 2.

Fig. 10. Time histories of observation vector with noise level of 10%

Fig. 10 shows the time histories of the measured variables with the noise level of 10%. Figs. 11 and 12 show the identification results of the dynamic responses and the track irregularities, respectively. As shown in Fig. 11, the differences between the estimated responses and the exact values slightly increase with the measurement noise level. The maximum relative errors of peak responses are 0.51%, 2.1% and 4.8% for the noise levels of 1%, 5% and 10%, respectively. Fig. 12 compares the identified and actual track irregularities. Similar observations were made. The identification errors at the peak location are 0.8%, 2.9% and 7.1% for the noise levels of 1%, 5% and 10%, respectively. In summary, the identification results of both the dynamic response and track irregularities are reasonably accurate, even though the measurement of the vehicle responses polluted by a high-level noise.

4.2.2. Effect of non-stationary running state In this case, the vehicle moves from the left end to the right end. To investigate the effect of vehicle’s non-stationary running state on the identified results, three scenarios of the running state are considered, i.e. Scenario I: velocity v = 220 km/h, acceleration a = 0 m/s2; Scenario II: velocity v = 190 km/h, acceleration a = 64  103km/h2; Scenario III: velocity v = 250 km/h, acceleration a = 64  103km/h2. In this numerical example, the observation vector y is established considering a measurement noise level of 5%, and the corresponding covariance matrices of noises for the three scenarios are determined as P = diag(4.91 m2/s4 0.23 rad2/s4)  106 , P = diag(5.91 m2/s4 0.30 rad2/s4)  10-6 and P = diag(5.09 m2/s4 0.20 rad2/s4)  10-6. The estimated responses and identified track irregularities with the three scenarios of running state are shown in Figs. 13–16. As shown in Fig. 13, the estimated responses of the vehicle and bridge agree well with the exact values in the stationary Scenario I. The maximum relative errors of the identified results are 2.1% and 1.9% for the vehicle response and the bridge response, respectively. Similar observations are also made from the identified track irregularities, as shown in Fig. 14. Fig. 15 shows the estimated responses of the vehicle and bridge in the non-stationary Scenarios II and III. It is noted that the responses of the Scenario II and the Scenario III are different, such as the peak value and the peak time. As shown in Fig. 15, the estimated responses of the vehicle and bridge agree well with the exact values in the two non-stationary Sce-

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

15

Fig. 11. Comparison of the exact and estimated velocity responses of the VB system (with different measurement noise levels).

Fig. 12. Comparison of the exact and identified track irregularities (with different measurement noise levels) (The blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Comparison of the exact and estimated velocity responses of the VB system in Scenario I.

16

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Fig. 14. Comparison of the exact and identified track irregularities in stationary running state (The blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Comparison of the exact and estimated velocity responses of the VB system in non-stationary running states (Scenarios II and III).

narios. Particularly, the track irregularities can be accurately identified by the proposed Kalman filter algorithm in both the Scenario II and III, as shown in Fig. 16. These numerical results indicate that the non-stationary running states have little influence on the identification results.

4.2.3. Effect of parameter uncertainties One of the major advantages of the proposed algorithm is that it uses the dynamic responses of operation vehicles for track irregularity identification. As a result, it is low cost and efficient compared with the expensive track inspection cars. However, it faces challenge that the dynamic parameters of both the operation vehicle and the bridge have uncertainty [35–37].The dynamic parameter uncertainty of a VB system includes the variations of the vehicle body mass mc, the vehicle stiffness k1,2, the vehicle damping c1,2, the bridge elastic modulus E, the bridge mass density mb, and the bridge damping ratio f. According to the references [23,35–37], the maximum deviations of the above parameters with respect to their exact values are up to 25%, 25%, 25%, 25%, 5% and 25% for mc, k1,2, c1,2, E, mb and f, respectively. Therefore, these parameters and the corresponding deviation levels are considered in this case to investigate the effects of parameter uncertainties on the identification results. In this case, a constant vehicle velocity of v = 250 km/h and a measurement noise level of 5% are considered. Two deviation levels of the parameters, i.e. 5% and 25% (except for mb), are taken into account. Figs. 17 and 18 shows the identification results of track irregularities for the two deviation levels, respectively. The identification errors at the peak location are 13.1%, 10.5%, 5.8%, 3.0%, 6.0% and 9.5% when considering a 5% deviation of k1,2, mc, E, f, c1,2 and mb, respectively. While the corresponding maximum absolute identification errors are 0.21 mm, 0.17 mm, 0.07 mm, 0.04 mm, 0.09 mm and 0.16 mm, respectively. Therefore, the comparison results shown in Fig. 17 indicates that the proposed Kalman algorithm can well identify the track irregularities when a low level of parameter uncertainty (5% deviation) is considered. However, the performance of the proposed track irregularity algorithm degrades if considering a high level of parameter uncertainty. Fig. 18 compares the identified and exact values of track irregularities when considering 25% deviation of the parameters, except for the bridge mass density. Numerical results reveal that maximum absolute identification errors over

17

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Fig. 16. Comparison of the exact and identified track irregularities in non-stationary running states (The blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)



Fig. 17. Comparison of the exact and identified track irregularities (parameter uncertainty with adeviation level of 5%, i.e. a. k1;2 ¼ ð1 - 5% Þk1;2 , b.        mc ¼ ð1 - 5% Þmc , c. E ¼ ð1 - 5% Þ E, d. f ¼ ð1 - 5% Þ f , e. c1;2 ¼ ð1 - 5% Þc 1;2 , f. mb ¼ ð1 - 5% Þmb , where k1;2 ,mc ,E,f ,c 1;2 and mb are the exact values; the blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the entire X-coordinate range are 0.48 mm, 0.40 mm, 0.42 mm, 0.09 mm and 0.22 mm, for k1,2, mc, E, f and c1,2, respectively. At the peak track irregularity location, the corresponding identification errors reach 30.0%, 24.4%, 25.5%, 5.4% and 13.1%, respectively. Consequently, the accuracy of the proposed algorithm mainly influenced by the deviation of the three parameters, namely, the vehicle body mass mc, the vehicle stiffness k1,2, and the bridge elastic modulus E. Therefore, to reduce the uncertainty level of these three parameters can significantly improve the identification accuracy of track irregularities. On the contrary, the effects of the uncertainties of the vehicle damping c1,2, the bridge damping ratio f and the bridge mass density mb are modest in general. 4.2.4. Effect of boundary springs The dynamic characteristics of bridges are usually affected by boundary conditions. This case considers a boundary spring kbs = 6.6  109Nm/rad that restricts the relative rotation at the junction of two spans (as shown in Fig. 19). As previously mentioned, a constant vehicle velocity of v = 250 km/h and a measurement noise level of 5% are considered.

18

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582



Fig. 18. Comparison of the exact and identified track irregularities (parameter uncertainty with a deviation level of 25%, as follows: a. k1;2 ¼ ð1  25% Þk1;2 ,         b. mc ¼ ð1  25% Þmc , c. E ¼ ð1  25% Þ E, d. f ¼ ð1 - 25% Þ f , e. c1;2 ¼ ð1 - 25% Þc 1;2 , where k1;2 ,mc ,E,f and c 1;2 are the exact values; the blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 19. Bridge model with a boundary spring.

Fig. 20 compares the first mode shapes with and without the boundary spring. It is observed that the mode shape with the boundary spring diverges slightly from the initial sinusoidal form. Moreover, the first natural frequency increases from 5.26 Hz to 6.46 Hz, which corresponds to a frequency increase of 22.8% compared with that of the without bounddary spring case. The estimated responses and identified track irregularities with the boundary spring are shown in Figs. 21 and 22, respectively. As shown in Fig. 21, the difference between the estimated responses and the exact values cannot be ignored, particularly for the velocity of the vehicle body. Similar performance degration can be observed in the identified track irregularities, as shown in Fig. 22. The maximum absolute identification errors over the entire X-coordinate range are 0.16 mm, and the identification error reaches 9.7% at the peak location. Therefore, modelling accurately the boundary conditions can improve the identification accuracy of track irregularities.

4.2.5. Comparative analysis of identification methods As previous mentioned, the conventional track irregularity identification approaches likely lead to large identification errors when applied to railway bridges, because the VB interaction cannot be considered. Additionally, the advancement of the proposed algorithm comparing with the conventional approaches requires a quantitative evaluation. To this end, a detailed comparison is conducted in this section. In this section, two popular track irregularity identification approaches, namely, the inertial reference method and the axle box acceleration method, are considered. In addition, the estimation results without considering the VB interaction are also presented. When neglecting the effect of the VB interaction, the presented Kalman filter algorithm is essentially similar to the method proposed by Tsunashima et al. [15]. If eliminating all submatrices or subvectors related to the generalized DOFs of bridges in the proposed numerical procedure (Eqs. (23), (26), (28), (31), (37), (39), (45)–(47)), the conventional inverse analyses method without VB interaction can also be simulated. In this case, a constant vehicle velocity of v = 250 km/h and a measurement noise level of 5% are considered. Fig. 23 shows the estimated vertical and pitching displacements of the vehicle body. It is observed that the estimated displacement responses considering the VB interaction match well with the exact values. However, if ignoring the VB interaction, the max-

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

19

Fig. 20. Comparison of the first mode shape with and without boundary spring.

Fig. 21. Comparison of the exact and estimated velocity responses of the VB system with the boundary spring.

Fig. 22. Comparison of the exact and identified track irregularities of the VB system with the boundary spring (the blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

imum peak estimation errors on the vertical and pitching displacements reach 50% and 196%, respectively. Therefore, neglecting the effect of the VB interaction leads to remarkable error in the response estimation. Fig. 24(a) and (b) compares the identified track irregularities with and without considering the effect of the VB interaction. As indicated in Fig. 24(a), the proposed algorithm when considering the VB interaction is capable of identifying the track irregularities on the bridge with high fidelity. The identification error at the peak location is only 4.1% and the maximum

20

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Fig. 23. Comparison of the exact and estimated responses of vehicle body with and without considering the VB interaction.

Fig. 24. Comparison of the track irregularities identified by different methods (the blue solid line represents the exact value, and the red dotted line represents the identified value). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

absolute identification error over the entire range is 0.06 mm. However, the estimation error is remarkable if neglecting the VB interaction effect, of which the identification error at the peak location is 357.2% and the maximum absolute identification error over the entire range is 2.50 mm. Fig. 24(c) and (d) show the identification results of the inertial reference method and the axle box acceleration method, respectively. The corresponding errors at the peak location can reach 135.9%. This happened at the time point when the front axle passed the midspan of the second span. On the other hand, the maximum absolute identification error is 1.36 mm for the both methods. It is worth noting that the two conventional methods overestimated the track irregularities, because the dynamic response amplification effect on the track inspection car due to the VB interaction cannot be eliminated. If the vehicle resonated with the bridge, such overestimation will be more serious. As a result, the identification errors of the two conventional methods are much larger than that of the proposed method in this paper. The comparison demonstrates that the proposed method outperforms the two conventional methods due to taking into account the VB interaction and its robustness of the algorithm itself. Fig. 25 shows the power spectral density (PSD) curves of the track irregularities versus the wavelength of track irregularity. We can observe that the VB interaction mainly affects the wavelength ranging from 5 m to 50 m (spatial frequency 0.2 m1 to 0.02 m1). Within the wavelength range of 5 m–50 m, the PSD without considering the VB interaction significantly diverges from the exact values, as indicated by an ellipse curve in Fig. 25. In contrast, the PSD curve matches well with the exact values when considering the VB interaction. Because the above wavelength range is of important in the railway track detection, the effect of VB interactions should be considered in track irregularity identification on bridges. For the

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

21

Fig. 25. Comparison of the exact and identified PSDs of track irregularities.

inertial reference method and the axle box acceleration method, the identification errors are also distributed within the wavelength range of 5–50 m (Fig. 25). These observations indicate that the identification results provided by the conventional methods are difficult to meet engineering requirements for track irregularity identification on railway bridges.

5. Discussions a) In this study, the rigid contact model is used for modeling the VB systems because of its high efficiency in computation. It is known that the elastic contact model [27–30] that takes the wheel-rail elastic contact and creep force into account is the other popular dynamic model for VB systems. The corresponding issues on track irregularity identification based on the elastic contact model deserve further study. b) It is noted that the estimation accuracy of track irregularities increases slightly with the increase of observed variables. € of the € and pitching acceleration h In this study, only two acceleration measurements, i.e. the vertical acceleration y c

vehicle body, are employed for track irregularity identification. If needed, the observed variables can also be increased using the proposed algorithm. For example, the relative displacement between the vehicle body and front wheel or rear wheel can be included in the observed vector. c) Numerical results in example 2 indicate that the identification errors caused by measurement noise, non-stationary running state, parameter uncertainties and the boundary spring effect are far less than those induced by neglecting VB interactions. This phenomenon stresses the important influence of the VB interaction on the track irregularity identification. Therefore, it is crucial to take the VB interaction into account for track irregularity identification on railway bridges.

6. Conclusion This paper presents a new Kalman filter algorithm for identifying the track irregularities on railway bridges using vehicle acceleration responses. The track irregularities are the excitations imposed on time-dependent VB systems, therefore, the track irregularity identification of railway bridges is essentially an inverse dynamic analysis. Due to the significant effect of the VB interaction on the system response, the VB interaction is considered in the proposed algorithm. Such track irregularity identification algorithm that considers the VB interaction is not available in the previous literature. Different from the conventional identification approaches, the proposed algorithm requires the information of bridge, i.e., the modal parameters. The numerical results in the example 1 demonstrate that the proposed algorithm only requires the information of the first several vibration modes of the railway bridge. When applied to the real-world railway bridges, the identification of the track irregularities is facing complex operating environment, such as the measurement noise, the non-stationary running states of vehicles, parameter uncertainties and the boundary condition errors. The numerical results in the example 2, i.e. a typical railway bridge in China, indicate that the proposed algorithm is capable of identifying the track irregularities using the on-board measurement signals polluted by high-level noise. In addition, the proposed algorithm is also effective for the three typical running states of the vehicle, i.e. constant velocity state, accelerated state and decelerated state. However, the parameter uncertainty of the dynamic models of the vehicle and the bridge generates significant negative effect on the track irregularity identification. To minimize the uncertainty on the model parameters and the model boundary condition is always of interest in engineering practice.

22

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

The numerical results also illustrate that the identification results significantly diverge from the exact values if ignoring the VB interaction. The proposed algorithm that considers the VB interaction offers a highly accurate identification of the track irregularities on railway bridges, which outperforms the conventional approaches. The real-time rail track monitoring considering the VB interaction will be put forwarded in the years ahead. CRediT authorship contribution statement X.Xiao: Methodology, formal analysis, validation, writing-original draft. W. Shen: Project administration, conceptualization, methodology, writing-review & editing. Z. Sun: Data curation, formal analysis. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant No.: 51838006), from the Natural Science Foundation of Hubei Province (Grant No.: 2018CFB429), from the Fundamental Research Funds for the Central Universities (HUST: 2018KFYYXJJ007), and from the Technology Innovation Special Projects of Hubei Province (Grant No.: 2017ACA183). Findings and opinions expressed here, however, are those of the authors alone, not necessarily the views of the sponsor. Appendix A. Subvector gbi and submatrix nbi The static axle’s load subvector gbi acting on the ith span is given by

8 gb1 ¼ wf N T1;x¼xf ; gb2 ¼ 0 0 < xf 6 Lc > > > > > > T T > Lc < xf 6 L > > gb1 ¼ wf N 1;x¼xf þ wr N 1;x¼xr ; gb2 ¼ 0 > > > > T < g ¼ wr N T L < xf 6 L þ Lc b1 1;x¼xr ; gb2 ¼ wf N 2;x¼xf > T T > > g ¼ 0; gb2 ¼ wf N 2;x¼xf þ wr N 2;x¼xr L þ Lc < xf 6 2L > > > b1 > > > T > 2L < xf 6 2L þ Lc > gb1 ¼ 0; gb2 ¼ wr N 2;x¼xr > > : gbi ¼ 0 ; gb2 ¼ 0 xf ¼ 0 or xf > 2L þ Lc

ðA1Þ

The input submatrix nbi of track irregularities on the ith span can be expressed as

h i 8 T > 0 < xf 6 Lc > > nb1 ¼ k1 N 1;x¼xf 0 ; nb2 ¼ 0 > > > h i > > > > nb1 ¼ k1 N T1;x¼xf  k2 N T1;x¼xr ; nb2 ¼ 0 L c < xf 6 L > > > > > h i h i > > < nb1 ¼ 0  k2 N T L < xf 6 L þ Lc ; nb2 ¼ k1 N T2;x¼xf 0 1;x¼xr h i > > T T > > L þ Lc < xf 6 2L > > nb1 ¼ 0 ; nb2 ¼ k1 N 2;x¼xf  k2 N 2;x¼xr > > > h i > > T > > 2L < xf 6 2L þ Lc > nb1 ¼ 0 ; nb2 ¼ 0  k2 N 2;x¼xr > > > : nb1 ¼ 0 ; nb2 ¼ 0 xf ¼ 0 or xf > 2L þ Lc

ðA2Þ

Appendix B. Matrices Hk, Gk and Dk of the observed equation The matrices Hk, Gk and Dk in the observed equation Eq. (31) are given by

2

ðL1 k1 þ L2 k2 Þ c H1   Hk ¼ 4 1 1 ðL1 k1 þ L2 k2 Þ Ic L21 k1  L22 k2 c H5 Ic " Dk ¼

 m1c ðk1 þ k2 Þ

k1 mc

k2 mc

L1 k 1 Ic

 L2Ick2

1 mc

c H2 c H6

 m1c ðc1 þ c2 Þ

ðL1 c1 þ L2 c2 Þ c H3   1 1 ðL1 c1 þ L2 c2 Þ Ic L21 c1  L22 c2 c H7 Ic 1 mc

c H4 c H8

3 5

ðA3Þ

# ;

Gk ¼ 0

ðA4Þ

23

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

where c H1 , c H2 , c H3 , c H4 , c H5 , c H6 , c H7 and c H8 can be expressed according to the following six scenarios. Scenario I: The moving distance xf of the vehicle satisfies 0 < xf  Lc

8 c H1 > > > > > > > < c H3 c H5 > > > > > c H7 > > : c H2

¼ m1c k1 N 1;x¼xf ¼ m1c c1 N 1;x¼xf ðA5Þ

¼ I1c L1 k1 N 1;x¼xf ¼ L1 c1 N 1;x¼xf 1 Ic

¼ c H4 ¼ c H6 ¼ c H8 ¼ 0

Scenario II: The moving distance xf of the vehicle satisfies Lc < xf  L

  8 > c H1 ¼ m1c k1 N 1;x¼xf þ k2 N 1;x¼xr > > >   > > > > þ c2 N 1;x¼xr c ¼ 1 c N > > < H3 mc 1 1;x¼xf  c H5 ¼ I1c L1 k1 N 1;x¼xf  L2 k2 N 1;x¼xr > > >   > > > > c H7 ¼ I1c L1 c1 N 1;x¼xf  L2 c2 N 1;x¼xr > > > : c H2 ¼ c H4 ¼ c H6 ¼ c H8 ¼ 0

ðA6Þ

Scenario III: The moving distance xf of the vehicle satisfies L < xf  L + Lc

8 > c H1 ¼ m1c k2 N 1;x¼xr ; c H2 ¼ m1c k1 N 2;x¼xf > > > > < c H3 ¼ 1 c2 N 1;x¼x ; cH4 ¼ 1 c1 N 2;x¼x r f mc mc

> c H5 ¼ I1c L1 k1 N 1;x¼xf ; c H6 ¼  I1c L2 k2 N 2;x¼xr > > > > :c ¼ 1 L c N ; c H8 ¼  I1c L2 c2 N 2;x¼xr H7 Ic 1 1 1;x¼xf

ðA7Þ

Scenario IV: The moving distance xf of the vehicle satisfies L + Lc < xf  2L

  8 1 > c H2 ¼ mc k1 N 2;x¼xf þ k2 N 2;x¼xr > > >   > > > > c H4 ¼ m1c c1 N 2;x¼xf þ c2 N 2;x¼xr > > <   c H5 ¼ I1c L1 k1 N 2;x¼xf  L2 k2 N 2;x¼xr > > >   > > >c ¼ 1 L c N > > H7 1 1 2;x¼xf  L2 c 2 N 2;x¼xr Ic > > : c H1 ¼ c H3 ¼ c H5 ¼ c H7 ¼ 0

ðA8Þ

Scenario V: The moving distance xf of the vehicle satisfies 2L < xf  2L + Lc

8 c H2 > > > > > > > < c H4 c H5 > > > > > c H7 > > : c H1

¼ m1c k2 N 2;x¼xr ¼ m1c c2 N 2;x¼xr ðA9Þ

¼  I1c L2 k2 N 2;x¼xr ¼

 I1c

L2 c2 N 2;x¼xr

¼ c H3 ¼ c H5 ¼ c H7 ¼ 0

Scenario VI: The moving distance xf of the vehicle satisfies xf = 0 or xf > 2L + Lc

c H1 ¼ c H2 ¼ c H3 ¼ c H4 ¼ c H5 ¼ c H6 ¼ cH7 ¼ c H8 ¼ 0

ðA10Þ

Appendix C. Nomenclature

Symbol

Description

yc hc ywf ywr v

Vertical displacement of the vehicle body Pitching displacement of the vehicle body Vertical displacement of the front wheel Vertical displacement of the rear wheel Velocity of the vehicle (continued on next page)

24

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

Nomenclature (continued) Symbol

Description

a mc Jc mwf mwr L1 L2 wf wr k1 c1 k2 c2 fbi Rf Rr Mbbi Cbbi Kbbi qbi Mvv Cvv Kvv pv p*bi Ni qn,i /n,i

Acceleration of the vehicle Mass of the vehicle body Mass moment of inertia of the vehicle body Mass of the front wheel Mass of the rear wheel Distances from the mass center of vehicle body to the front axle Distances from the mass center of vehicle body to the rear axle Static load of the front axle Static load of the rear axle Stiffness of the spring connection at the front axle Damping coefficient of the damper connection at the rear axle Stiffness of the spring connection at the rear axle Damping coefficient of the damper connection at the front axle Dynamic loads acting on the ith bridge-span due to the moving vehicle Track irregularity at the front wheel Track irregularity at the rear wheel Mass matrix of the ith bridge span Damping matrix of the ith bridge span Stiffness matrix of the ith bridge span generalized DOF vector of the ith bridge span Mass matrix of the vehicle Damping matrix of the vehicle Stiffness matrix of the vehicle Load vector of the vehicle in the VB system Load vector of the ith bridge span in the VB system Shape function vector of the ith bridge span The nth order modal coordinate of the ith bridge span The nth mode shape function of the ith bridge span The nth mode frequency of each span The nth mode damping ratio of each span Total mode number used in the numerical calculations X-coordinate Mass per unit length of the bridge Elastic modulus of the bridge Moment of inertia of the bridge cross section Dynamic load acting on the bridge at the front axle Dynamic loads acting on the bridge at the rear axle X-coordinates of the front wheel, moving distance of the vehicle X-coordinates of the rear wheel Distance between the front and rear axles Length of each span Mass matrix of the ith bridge span induced by the VB interaction Damping matrix of the ith bridge span induced by the VB interaction Stiffness matrix of the ith bridge span induced by the VB interaction Vehicle-bridge interaction damping matrix induced by the VB interaction Vehicle-bridge interaction stiffness matrix induced by the VB interaction Static axle’s load vector Generalized DOF vector of the VB system Mass matrix of the VB system Damping matrix of the VB system Stiffness matrix of the VB system load vector of the VB system Track irregularity vector of the VB system Input matrix of track irregularities of the VB system Static axle’s load subvector acting on the ith span Input submatrix of track irregularities on the ith span

xn fn m x mb E I pf pr xf xr Lc L M*bbi C*bbi K*bbi C*vbi, C*biv K*vbi, K*biv f q M C K p R F

gbi nbi

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

25

Nomenclature (continued) Symbol

Description

x A B E t Dt k

State vector of the VB system State transformation coefficient matrix of the continuous-time state equation Input matrix of known load of the continuous-time state equation Input matrix of unknown load of the continuous-time state equation Time Time step Number of time step State transformation coefficient matrix of the discrete-time state equation Input matrix of known load of the discrete-time state equation Input matrix of unknown load of the discrete-time state equation Covariance of the system noise System noise vector Observed vector Acceleration output matrix Velocity output matrix Displacement output matrix Track irregularity output matrix State output matrix of the observed equation Output matrix of known load of the observed equation Output matrix of unknown load of the observed equation Measurement noise vector Covariance of the measurement noise Optimal estimation of the state vector Predicted state vector

U U

X Q w y Ca Cv Cd Cr H G D v P ^ x 

x ^

e  e

Optimal estimation of the error vector

L L* ^ P

Coefficient matrix for the optimal estimation of the state vector Gain matrix of the Kalman estimator Optimal estimation of the covariance matrix

P C ^0 x ^0 P

Predicted covariance matrix

W R I

Weight matrix Optimal estimation of track irregularity vector Identity matrix



Predicted error vector

Coefficient matrix for predicting the covariance matrix Initial value of the optimal estimated state vector Initial value of the optimal estimated covariance matrix

References [1] S. Bruni, R.M. Goodall, T.X. Mei, H. Tsunashima, Control and monitoring for railway vehicle dynamics, Veh. Syst. Dyn. 45 (7–8) (2007) 765–771. [2] P.E. Waston, C.S. Ling, C.J. Goodman, P. Li, R.M. Goodall, Monitoring vertical track irregularity from in-service railway vehicles, J. Rail Rapid Transit. 221 (1) (2007) 75–88. [3] S. Alfi, S. Bruni, Estimation of long wavelength track irregularity from on board measurement, in: Proc. 4th IET int. conference on railway condition monitoring, Derby, UK., 2008. [4] Y.B. Yang, J.D. Yau, Z. Yao, Y.S. Wu, Vehicle-Bridge Interaction Dynamics: With Applications to High-Speed Railways, World Scientific, 2004. [5] Y.S. Cheng, F.T.K. Au, Y.K. Cheung, Vibration of railway bridges under a moving train by using bridge-track-vehicle element, Eng. Struct. 23 (2001) 1597–1606. [6] P. Lou, Z.W. Yu, F.T.K. Au, Rail-bridge coupling element of unequal lengths for analyzing train-track-bridge interaction systems, Appl. Math. Model. 36 (2012) 1395–1414. [7] A. Haigermoser, B. Luber, J. Rauh, G. Gräfe, Road and track irregularities: measurement, assessment and simulation, Veh. Syst. Dyn. 53 (7) (2015) 878– 957. [8] M. Molodova, Z. Li, R. Dollevoet, Axle box acceleration: Measurement and simulation for detection of short track defects, Wear 271 (1–2) (2011) 349– 356. [9] Y. Naganuma, M. Kobayashi, M. Nakagawa, T. Okumura, Condition monitoring of shinkansen tracks using commercial trains, in: Proc. 4th IET Int. Conference on Railway Condition Monitoring, Derby, UK., (2008) 1-6. [10] H. Tsunashima, Y. Naganuma, A. Matsumoto, et al, Condition monitoring of railway track using in-service vehicle, Reliabil. Saf. Railway 12 (2012) 334– 356.

26

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

[11] J.S. Lee, S. Choi, S.S. Kim, C. Park, Y.G. Kim, A Mixed Filtering Approach for Track Condition Monitoring Using Accelerometers on the Axle Box and Bogie, IEEE Trans. Instrum. Meas. 61 (3) (2012) 749–758. [12] J. Real, P. Salvador, L. Montalbán, M. Bueno, Determination of rail vertical profile through inertial methods, J. Rail Rapid Transit 225 (1) (2011) 14–23. [13] T. Karis, M. Berg, S. Stichel, M. Li, D. Thomas, B. Dirks, Correlation of track irregularities and vehicle responses based on measured data, Veh. Syst. Dyn. 56 (6) (2018) 967–981. [14] P. Czop, K. Mendrok, T. Uhl, Application of inverse linear parametric models in the identification of rail track irregularities, Arch. Appl. Mech. 81 (11) (2011) 1541–1554. [15] H. Tsunashima, Y. Naganuma, T. Kobayashi, Track geometry estimation from car-body vibration, Veh. Syst. Dyn. 52 (2014) 207–219. [16] R. Schenkendorf, B. Dutschk, K. Lüddecke, J.C. Groos, Improved Railway Track Irregularities Classification by a Model Inversion Approach, in: Proc. 3th European Conference of the Prognostics and Health Management Society, 2016. [17] H.P. Zhu, L. Ye, S. Weng, W. Tian, Damage identification of vehicle-track coupling system from dynamic responses of moving vehicles, Smart Struct. Syst. 21 (5) (2018) 677–686. [18] C. Li, S. Luo, C. Cole, M. Spiryagin, An overview: modern techniques for railway vehicle on-board health monitoring systems, Veh. Syst. Dyn. 55 (2017) 1045–1070. [19] M. Majka, M. Hartnett, Dynamic response of bridges to moving trains: A study on effects of random track irregularities and bridge skewness, Comput. Struct. 87 (2009) 1233–1252. [20] X. Xiao, Y. Yan, B. Chen, Stochastic dynamic analysis for vehicle-track-bridge system based on probability density evolution method, Eng. Struct. 188 (2019) 745–761. [21] Y.B. Yang, J.P. Yang, State-of-the-art review on modal identification and damage detection of bridges by moving test vehicles, Int. J. Struct. Stab. Dyn. 18 (02) (2018) 23. [22] C.S. Cai, J.X. Hu, S.R. Chen, Y. Han, W. Zhang, X. Kong, A coupled wind-vehicle-bridge system and its applications: a review, Wind Struct. 20 (2) (2015) 117–142. [23] H.P. Wan, Y.Q. Ni, An efficient approach for dynamic global sensitivity analysis of stochastic train-track-bridge system, Mech. Syst. Signal Process. 117 (2019) 843–861. [24] H. Xia, N. Zhang, Dynamic analysis of railway bridge under high-speed trains, Comput. Struct. 83 (2005) 1891–1901. [25] X. Xiao, W.X. Ren, A Versatile 3D vehicle-track-bridge element for dynamic analysis of the railway bridges under moving train loads, Int. J. Struct. Stab. Dyn. 19 (4) (2019) 1950050. [26] W.M. Zhai, Z. Han, Z. Chen, Train–track–bridge dynamic interaction: a state-of-the-art review, Veh. Syst. Dyn. 57 (7) (2019) 984–1027. [27] W.M. Zhai, H. Xia, C.B. Cai, et al, High-speed train-track-bridge dynamic interactions Part I: theoretical model and numerical simulation, Int. J. Rail Trans. 1 (1–2) (2013) 3–24. [28] Z.B. Jin, S. Pei, X. Li, et al, Effect of vertical ground motion on earthquake-induced derailment of railway vehicles over simply-supported bridges, J. Sound Vib. 383 (2016) 277–294. [29] S.C. Yang, S.H. Hwang, Train-track-bridge interaction by coupling direct stiffness method and mode superposition method, J. Bridge Eng. 21 (10) (2016) 04016058. [30] Y.S. Lee, S.H. Kim, J. Jung, Three-dimensional finite element analysis model of high-speed train-track-bridge dynamic interactions, Adv. Struct. Eng. 8 (5) (2005) 513–528. [31] Y.B. Yang, J.D. Yau, Y.S. Wu, Vehicle-Bridge Interaction Dynamics: With Applications to High-Speed Railways, World Scientific Publishing Co., Pte. Ltd., Singapore, 2004. [32] Q. Zeng, E.G. Dimitrakopoulos, Vehicle-bridge interaction analysis modeling derailment during earthquakes, Nonlinear Dynam. 93 (4) (2018) 2315– 2337. [33] P. Lou, Z.W. Yu, F.T.K. Au, Rail-bridge coupling element of unequal lengths for analyzing train–track–bridge interaction systems, Appl Math Model. 36 (2012) 1395–1414. [34] Q. Zeng, C.D. Stoura, E.G. Dimitrakopoulos, A localized lagrange multipliers approach for the problem of vehicle-bridge-interaction, Eng. Struct. 168 (2018) 82–92. [35] Z.W. Yu, J.F. Mao, F.Q. Guo, Non-stationary random vibration analysis of a 3D train–bridge system using the probability density evolution method, J. Sound Vib. 366 (2016) 173–189. [36] J.F. Mao, Z.W. Yu, Y.J. Xiao, Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method, Probab. Eng. Mech. 46 (2016) 48–61. [37] L. Xu, W. Zhai, A novel model for determining the amplitude-wavelength limits of track irregularities accompanied by a reliability assessment in railway vehicle-track dynamics, Mech. Syst. Signal Process 86 (2017) 260–277. [38] Y.B. Yang, J.D. Yau, L.C. Hsu, Vibration of simple beams due to trains moving at high speeds, Eng. Struct. 19 (11) (1997) 936–944. [39] Y. Lei, Y. Jiang, Z. Xu, Structural damage detection with limited input and output measurement signals, Mech. Syst. Signal Process. 28 (2012) 229–243. [40] Y. Lei, D. Xia, K. Erazo, S. Nagarajaiah, Structural damage detection with limited input and output measurement signals, Mech. Syst. Signal Process. 127 (2019) 120–135. [41] H. Ebrahimian, R. Astroza, J.P. Conte, Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method, Earthq. Eng. Struct. Dyn. 44 (10) (2015) 1495–1522. [42] S.E. Azam, E. Chatzi, C. Papadimitriou, A dual Kalman filter approach for state estimation via output-only acceleration measurements, Mech. Syst. Signal Process. 60 (2015) 866–886. [43] Z. Lai, Y. Lei, S. Zhu, Y.L. Xu, X.H. Zhang, S. Krishnaswamy, Moving-window extended Kalman filter for structural damage detection with unknown process and measurement noises, Measurement 88 (2016) 428–440. [44] M.N. Chatzis, E.N. Chatzi, S.P. Triantafyllou, A discontinuous extended Kalman filter for non-smooth dynamic problems, Mech. Syst. Signal Process. 92 (2017) 13–29. [45] E. Lourens, E. Reynders, G. De Roeck, G. Degrande, G. Lombaert, An augmented Kalman filter for force identification in structural dynamics, Mech. Syst. Signal Process. 27 (2012) 446–460. [46] Y. Lei, D.T. Wu, S.Z. Lin, Integration of decentralized structural control and the identification of unknown inputs for tall shear building models under unknown earthquake excitation, Eng. Struct. 52 (2013) 306–316. [47] Y. Lei, D. Xia, K. Erazo, S. Nagarajaiah, A novel unscented Kalman filter for recursive state-input-system identification of nonlinear systems, Mech. Syst. Signal Process. 127 (2019) 120–135. [48] L. Guo, Y. Ding, Z. Wang, G. Xu, B. Wu, A dynamic load estimation method for nonlinear structures with unscented Kalman filter, Mech. Syst. Signal Process. 101 (2018) 254–273. [49] R. Astroza, H. Ebrahimian, Y. Li, J.P. Conte, Bayesian nonlinear structural FE model and seismic input identification for damage assessment of civil structures, Mech. Syst. Signal Process. 93 (2017) 661–687. [50] R. Astroza, H. Ebrahimian, J.P. Conte, Performance comparison of Kalman-based filters for nonlinear structural finite element model updating, J. Sound Vib. 438 (2019) 520–542. [51] A. Calabrese, S. Strano, M. Terzo, Adaptive constrained unscented Kalman filtering for real-time nonlinear structural system identification, Struct. Control. Hlth. 25 (2) (2018) e2084. [52] R. Astroza, A. Alessandri, J.P. Conte, A dual adaptive filtering approach for nonlinear finite element model updating accounting for modeling uncertainty, Mech. Syst. Signal Process. 115 (2019) 782–800.

X. Xiao et al. / Mechanical Systems and Signal Processing 138 (2020) 106582

27

[53] R. Astroza, A. Alessandri, Effects of model uncertainty in nonlinear structural finite element model updating by numerical simulation of building structures, Struct. Control. Hlth. 26 (3) (2019) e2297. [54] R.W. Clough, J. Penzien, D.S. Griffin, Dynamics of Structures, McGraw-Hill, 1993. [55] L. Fry´ba, Vibration of solids and structures under moving loads, 3rd Edition, Academia, Prague, 1999. [56] K. Liu, E. Reynders, G. De Roeck, et al, Experimental and numerical analysis of a composite bridge for high-speed trains, J. Sound Vib. 320 (1–2) (2009) 201–220. [57] M.S. Allen, T.G. Carne, Delayed, multi-step inverse structural filter for robust force identification, Mech. Syst. Signal Process. 22 (5) (2008) 1036–1054. [58] Y. Liu, T. Li, Y.Y. Yang, et al, Estimation of tire-road friction coefficient based on combined APF-IEKF and iteration algorithm, Mech. Syst. Signal Process. 88 (2017) 25–35. [59] K. Soal, Y. Govers, J. Bienert, et al, System identification and tracking using a statistical model and a Kalman filter, Mech. Syst. Signal Process. 133 (2019) 106127. [60] S. Strano, M. Terzo, On the real-time estimation of the wheel-rail contact force by means of a new nonlinear estimator design model, Mech. Syst. Signal Process. 105 (2018) 391–403. [61] V. Niola, S. Strano, M. Terzo, A random walk model approach for the wheel-rail contact force estimation, J. Dyn. Syst-t. Asme. 140 (7) (2018) 071016. [62] S. Strano, M. Terzo, Review on model-based methods for on-board condition monitoring in railway vehicle dynamics, Adv. Mech. Eng. 11 (2) (2019), 168781401982679. [63] R.E. Kalman, R.S. Bucy, New results in linear filtering and prediction theory, J. Basic Eng. 83 (1) (1961) 95–108. [64] C.K. Chui, G. Chen, Kalman Filtering, Springer International Publishing, 2017.