Dynamic Numerical Simulation Model and Time-Domain Identification of Flexible AERORail Structure

. A simplifed theoretical model of the AERORail vehicle-bridge coupling system and a corresponding numerical simulation system in Simulink are established. Based on several widely used methods for modelling and simplifying vehicle systems, the Simulink simulation system used in this study, including the vehicle system and the bridge (AERORail) system, is presented. Identifcation examples using a moving load model and a simplifed 1/4-scale vehicle model are established. Te simulation results agree with the data of the simplifed dynamic model, with errors between 2.9% and 4.72%, and a satisfactory accuracy is achieved even for single-point signal identifcation, thereby verifying the correctness of the simplifed dynamic model of the AERORail system and the improved time-domain method based on the method of moments.


Introduction
Te AERORail transportation system, referred to as AERORail, is a new type of cable-rail composite structure consisting of multispan continuous chord cables, rails, lower supports, and related power and control systems [1] and is somewhat similar in appearance to existing cable-supported suspension bridges ( Figure 1). However, in contrast to traditional bridges, the AERORail structure uses steel as the structural material instead of concrete and prestressed concrete (PC) for the superstructure, eliminates the heavy bridge deck in traditional bridges, and forms a spatial forceresisting system by directly using the steel rail and the chords together. Te structural stifness is mainly derived from the stress stifness formed by the pretensioning of the chord cables, and the system exhibits the obvious characteristics of a fexible structure.
Traditional light rail systems often require the construction of heavy girders to carry the rail and vehicles, such as the PC rail girders and steel box rail girders used in the Chongqing light rail [2], the PC channel girders used in the Shanghai rail transit system, and the German-made suspended sky train planned for introduction in Ningbo [3].
Compared with the abovementioned traditional structures, AERORails have the advantages of a simple structural form, low construction cost, less resource consumption, fast construction speed, and low impact on the environment. In view of the large proportion of steel structure in AERORails and the need for concrete and earthwork only for the foundation, these systems can be constructed at a fast pace with relatively little pollution to the environment. Moreover, the superstructure of an AERORail occupies only a small space, and its structure is completely hollowed out and easy to see through; hence, it has little impact on the natural or urban landscape. Terefore, AERORails have signifcant advantages in terms of environmental protection.
Research on the AERORail structure has mainly focused on its static and dynamic behaviour. Researchers at Tongji University preliminarily verifed the feasibility of the AERORail structure using 1 : 20 and 1 : 15 scale models and used a virtual prototype to investigate the dynamic and static behaviour of the structure [4,5]. Tey conducted numerical analysis on the static defections of AERORails with diferent spans under diferent loads and pretensions and experimentally verifed the numerical results using a 1 : 1 full-scale AERORail model. On this basis, they investigated the dynamic defection responses of AERORails with diferent spans using vehicle-bridge coupling theory.
Studies have demonstrated that the cable stress increment, dynamic defection, and structural stifness of the AERORail structure do not change signifcantly under lowspeed moving loads, that the structural stifness increases signifcantly with increasing cable force and midspan support height, and that there is a nonlinear relationship between vehicle speed and dynamic defection [4,5]. Although these studies have preliminarily revealed some of the static and dynamic characteristics of the AERORail structure, their results are still inadequate for the further development of related research and engineering applications.
In practical engineering, the consideration and selection of loads are the frst and also one of the most important steps in designing a structure. Static loads are often determined based on statistical data from feld investigations and using a safety factor to ensure a certain level of structural performance [6]. Dynamic loads, including moving loads and dynamic loads in the narrow sense, come from efects of the external environment on the structure, such as wind, an earthquake, or an explosion. Tese efects are collectively referred to as the external excitation. Te dynamic response of a structure, including its acceleration, velocity, and displacement, is the result of the interaction between the structure's own properties and the external excitation. In some cases, the structure has little efect on the excitation (e.g., earthquake), but there are many cases of coupling (e.g., wind-induced vibration and wave action), for which the research often has a high degree of complexity. Compared to the traditional problem of fnding the vibrational response with known structural parameters and external excitations, the study of excitation, especially the calculation of excitation based on existing responses and structural parameters, is an inverse problem that emerged late in the study of vibration systems, i.e., the dynamic load identifcation problem.
Dynamic load identifcation technology was initially developed in the 1970s and was frst used for military purposes. Currently, the mainstream dynamic load identifcation methods include time-domain methods (TDMs) and frequency-domain methods (FDMs). FDMs are based on the Fourier transform or the Laplace transform. By establishing a transformation relationship between the excitation and response in the frequency domain (such as the frequency response function), the corresponding excitation can be calculated from the measured response. FDMs were studied earlier and are relatively mature, but they impose a requirement on the sample length of the response signal (it must usually be longer than a specifed length). Terefore, FDMs are generally used only for stationary random loads or steady-state dynamic loads [7]. TDMs are usually based on the kinetic equations of the system. By inverting the convolution integral of structural responses, the dynamic load applied to the structure is determined throughout the time. Te responses used in identifcation can be displacement, velocity, and acceleration. Some classic methods in the time domain are the deconvolution method, weighted acceleration method, and function ftting method [8,9]. In recent years, a series of new methods have also been reported, such as time fnite element methods, inverse system methods, neural network-based methods, and transformation with orthogonal wavelet operators [10]. In contrast to FDMs, TDMs are sensitive to boundary conditions and initial values, but they can be used for nonlinear systems and to identify transient impact loads [11,12], and their identifcation accuracy is not infuenced by the signal acquisition method [13,14]. Additionally, TDMs can capture the actual history of time-dependent loads at each specifc time increment. Tis history of load usually has a clear physical meaning and is therefore more practical to guide the design and improvement of AERORail structure.
In bridge engineering, there is another branch of dynamic load identifcation-moving load identifcation. In contrast to general dynamic load identifcation, moving load identifcation mostly involves vehicles traveling on bridges. It is very important to determine the dynamic efect of moving vehicles on the structure in bridge design and construction. However, because the interaction between the two is closely related to the design parameters of the bridge, the design parameters of the vehicles, and the driving speed, the force between each vehicle and the bridge is often diffcult to quantify accurately. Terefore, it is necessary to calculate the force between each vehicle and the bridge using the dynamic load identifcation method [15]. A very prominent feature of moving load identifcation is that the action point of the external excitation changes with time. As a result, some traditional methods for dynamic load identifcation, such as frequency response function inversion, cannot be directly applied to moving load identifcation. Based on the advantages of TDMs, we use a time-domain method which is improved by the method of moment to identify the contact force between the vehicle and the structure.
To verify whether the improved time-domain method (ITDM) can capture the dynamic load applied on the structure, a simplifed structure-vehicle model is used in the test case. Tis simplifed model is a system of partial differential equations (ODE) containing both the vehicle and the structure's kinetic equations, where the structural and material parameters of AERORail are utilized. Te solution of this system is obtained by Simulink.
Te article is structured as follows: Section 2 proposes a system of ODEs as a simplifed structure-vehicle model, which is used in the next section for verifcation. Te numerical procedures to solve the equations are also introduced therein. Te theory and numerical implementation of ITDM are briefy explained in Section 3. To fully investigate ITDM's ability of capturing the dynamic external load, two test cases with a moving force and a ¼ vehicle model are presented in Section 4. Te summary of main conclusions of this study is provided in Section 5.

Simplified Dynamic Numerical Model of an AERORail System
Te joint vibration efect of vehicle and bridge structures, i.e., vehicle-bridge coupled vibration, is an important property in bridge structural dynamics. In engineering practice, usually due to the limitations of the calculation method and time cost, only the vibration of the bridge itself is considered, or each vehicle is simplifed as a moving mass block in structural analysis and verifcation, which clearly fails to fully and truly refect the efect of vehicle-bridge coupled vibration. Generally, to comprehensively consider the vehicle-bridge coupled vibration problem, the vehicle dynamics, rail dynamics, and wheel-rail contact relationship should be considered and analysed as a whole coupled system. To this end, Hwang and Nowak [16], Wang and Huang [17], Yang et al. [18,19], Tan et al. [20], and Kwasniewski et al. [21] proposed numerical vehicle-bridge coupling models based on beam elements, plane bar systems, or solid simulation models, which gradually achieved results with satisfactory accuracy and were relatively consistent with experimental results. In these vehicle-bridge coupling models, there are four vehicle model options ( generates accurate results and can represent the vibration efect caused by "swaying motions" and "nodding motions," but its calculation is complex and costly.
Notice that in the rest of the paper, the coordinate system of the beam is along its axis with the origin at the end where the load/vehicle makes the frst contact with the beam.
Tere are some assumptions shared by these vehicle models: (1) Te friction force between wheels and the structure is ignored. Tis is mainly because the defection is a major concern in structural safety; therefore, the vertical load is what engineers care more about. (2) Te kinematics of vehicle systems does not take ambient vibrations (for example, vibration from the motor) into account. (3) Te parameters assigned to the systems are from numerical analysis or an investigation of the literature. Tere could be the diference between these chosen parameters and the real properties of the structure.
For the dynamic load identifcation problem considered in this study, the full vehicle model and the 1/2-scale vehicle model are too complex, the 1/4-scale vehicle model is appropriate, and the moving mass model is computationally simple and can be used as a simplifed model for practical calculation. In the following, the simplifed 1/4-scale vehicle model and the moving load (ML) model are used to calculate the vehicle-bridge coupled vibration of the AERORail structure using modal decomposition. Ten, this numerical simulation system is used to verify the improved TDM (ITDM) based on the method of moments.

Simplifed Dynamic Numerical Model of the AERORail
Structure. It is assumed that a vehicle moving on the AERORail structure can be simplifed as a vibration system composed of masses, springs, and dampers. Taking the single-degree freedom system as an example, the vibration equation can be expressed as the following equation: where m c , c c , and k c are the mass, damping, and contact stifness, respectively, of the simplifed vehicle system; w c is the vertical displacement of the centre of mass of the simplifed vehicle system relative to the geodetic coordinate system; w(ct) is the defection of the bridge at the coordinate ct at time t, where c is the moving speed of the vehicle system; f ct is the contact force between the vehicle system and the bridge system; and m c g is the gravitational force on Shock and Vibration 3 the vehicle system. Te corresponding vibration equation of the bridge structure can be expressed as the following equation: where q n is the modal displacement, ξ n is the modal damping ratio, ω n is the modal circle frequency, ρ b b is the linear mass of the beam, L b is span, and n is the order. Notice that the bridge structure here is assumed to be an Euler-Bernoulli beam. Te above two equations and their parameters together form a simplifed dynamic numerical model of the AERORail. Te vibration diferential equations of this coupled system can generally be solved using the Newmarkβ and Wilson-θ methods. In this study, Simulink is used to build a simulation system of the model, and the ode45 solver is employed to solve the equations using the 4/5th-order Runge-Kutta method.

Calculation Module of the Vehicle System.
To build a calculation module using Simulink, it is necessary to frst change the vibration diferential equation into a form that can be directly calculated by the integrator, that is, an explicit expression of acceleration. By slightly rearranging terms in equation (1), we obtain Te constant parameters in equation (3) that must be determined in advance are ξ c , ω c , m c (or m c , k c , and ξ c ), c and g, where g is the acceleration of gravity. Te defection of the contact point, w(ct), must be "input" from the outside continuously to calculate the contact force, f ct , which is "output" continuously. Te calculation module of the vehicle system in Simulink is shown in Figure 3.
To make the vehicle system enter the bridge structure "smoothly," it is necessary to eliminate the transient vibration of the vehicle system caused by gravity at the beginning of the calculation. To this end, the following initial integration condition is introduced into the integrator: To simulate multivehicle or multiwheels conditions, it is only necessary to make multiple copies of this module and connect them to the calculation process.
Te contact force in Figure 3 is an interface to the AERORail calculation module, in which the location and the value of the load are the current coordinate and the contact force of the vehicle. Notice that the contact force itself as an unknown is solved together with other variables in the Simulink using an explicit solver.

Calculation Module of the AERORail System.
Te modal acceleration of the bridge structure, i.e., the AERORail structure, is expressed as follows: where the constant input parameters are ρ b , L b , c, ξ n , and ω n .
Te contact force f ct is obtained from the vehicle system calculation module. Te single-mode vibration calculation module of the AERORail system in Simulink is shown in Figure 4. Te module in Figure 4 is a single-mode vibration calculation module. Because the multimode vibration response must be calculated in the simulation, it is necessary to  Te locs and f (t) in the AERORail calculation module are the current coordinates and values of the excitation force. For example, when vehicle module is linked to the system, locs1 and f (t) 1 are the coordinates and contact force (see Figure 3) of the frst ¼ vehicle model. Te vehicle's coordinates are taken care of by an overall control unit which evaluates locs1 � ct at every simulation instance. Moreover, in the moving load example, the f(t)1 is not an unknown but a known function predefned by the user.
Since the structural modes are used in the ODE, the structural responses at certain points, such as defections, speeds, and accelerations, are recovered from the simulation results using the amplitudes of all structural modes involved in the simulation.

ITDM Based on the Method of Moments (MoM)
TDMs for identifcation exhibit instability in practical identifcation, which is mainly refected in the large size and severe ill-conditioning of the solution matrix [22][23][24]. In the area of antennas, microwaves, and electromagnetic waves, the method of moments is a widely accepted tool to address this issue. As a long-established procedure in the statistics, the method of moments provides a good estimation that matches the true and sample moments. Since the mathematics of inversing the convolution integral equations are replaced by statistic estimation, the algorithm's stability and the solution's quality are signifcantly improved. We present here a brief introduction to the MOM [A] for an inhomogeneous equation as follows: with L, g, and f being the linear operator, a known forcing function, and an unknown function, respectively. Given a set of basis functions f n that are nearly independent, an approximate solution of f can be expressed as follows: f � N n�1 a n f n , where a n are weighting coefcients to be determined. In a Hilbert space, the inner product of functions f and g is defned as follows:

Shock and Vibration
Te solution to equation (5) is obtained when the inner product of residual ε n and M selected test functions q m equals zero.  Equation (9) can be written in the following matrix form: 6 Shock and Vibration Te original TDM is to solve the following equations at N sampling instances of acceleration: where € h n (k) is the unit impulse response incurred by excitation at k-th sampling instance corresponding to the n-th mode with damping, A n is a coefcient matrix corresponding to n-th structural mode, N c is the number of sampling points to the moving force, and € w is the column of accelerations samples.
Using Legendre polynomials p i up to the N p -th order as a set of orthogonal basis functions yields equation (13):

Shock and Vibration
Using the Dirac function δ(t − t j ) as the test function, the resultant discrete equations of acceleration and excitation force read [25]: Equation (9) is the system of linear equations for the ITDM based on the method of moments. Let Ten, equation (9) can be simplifed as follows: Solving it using the least squares method yields the following equation: An approximate solution of the moving load is obtained by substituting equations (16) into (12).
In order to investigate the weight of Legendre polynomials of diferent orders, we plot the values of α i in equation (15) obtained from single-point and multipoint identifcations in Figure 5.
In fact, α approaches zero when N p is greater than a threshold value. Te physical meaning of this phenomenon is that in practical calculations, using Legendre polynomials beyond a threshold order has a little infuence on the calculation results. Empirically choosing an appropriate order N p of Legendre polynomials helps improve the computational efciency without sacrifcing computational accuracy.

Simplified Dynamic Model of AERORail and Examples of TDMs for Identification
To verify the correctness of the ITDM based on the method of moments and to ensure that the two theories are consistent and the calculations are properly connected, two identifcation examples are given in this section with moving loads (ML) and a simplifed 1/4-scale vehicle model [26]. Moving loads model is probably the simplest case in the realm of load identifcation. However, it is very useful when validating algorithms and corresponding implementations, since the input is clearly defned. Te ¼-scale vehicle model is adopted to test the method's ability in a close-to-reality scenario, where the structure-vehicle interaction is the dominant efect.
Te calculation process and results of these two examples are aforementioned.

Example with the ML Model.
Te parameters of the simply supported beam simulated in this example are shown in Table 1. Te listed parameters are based on the numerical estimation of an under-construction AERORail structure.
Te ML is expressed as follows: Te ML p(t)-t curve is plotted in Figure 6. By substituting the load and parameters into the simplifed dynamic model of the AERORail, the accelerations at the 1/4, 1/2, and 3/4 spans of the beam and the defection at the 1/2 span of the beam are calculated, and the results are shown in Figures 7 and 8.
Te acceleration curves indicate that the acceleration oscillation of the beam gradually increases as the moving load moves and gradually decreases after the load approximately passes the midspan. Te defection increases to the maximum when the moving load moves near the midspan and then gradually returns to the equilibrium position and performs free damped vibration. Tis calculation result is close to the actual engineering experience, indicating that the simplifed dynamic numerical model of the AERORail can meet the calculation requirements to a certain extent.
Te acceleration time history of the beam at the 1/2 span is used as the response sample. Te signal is resampled with a frequency of f s � 200 Hz and input together with the parameters of the simply supported beam of the ML model in Table 1 into the calculation program of the ITDM based on the method of moments. Te calculation fow of the program is shown in Figure 9.  Parameters Legendre polynomials of order m � 100 are used in the identifcation calculation to obtain the identifed moving load p(t)-t curve, as shown in Figure 10 Figure 11: Time history of acceleration at diferent spans using the vehicle system model. [27]. Te larger the condition number is, the closer the eigenvalue is to zero and the more severe the ill-conditioning of the matrix. To verify that the ITDM is superior to the original TDM, the matrix condition numbers in the cases of single-point identifcation and multipoint identifcation are calculated and listed in Table 2. In 0, it was shown that the ITDM efectively reduces the matrix condition number and improves the matrix state, making the linear equations easier to solve.

Example with a Simplifed 1/4-Scale Vehicle System.
Te parameters of the simply supported beam in this example are the same as those in the above example. Te parameters of the vehicle system are consistently selected as follows: m c � 5000 kg, k c � 100000 N/m, and ξ c � 0.15. Te accelerations at the 1/4, 1/2, and 3/4 spans of the beam and the defection at the 1/2 span of the beam are calculated, and the results are shown in Figures 11 and 12.
Te identifcation results are shown in Figure 13. Te ITDM based on the method of moments performs reasonably well in identifying the vehicle system model, with only a small error (∼2.9%) in the single-point recognition when the vehicle travels to the midspan. Tis example demonstrates that the ITDM is also applicable to systems with a bridge-vehicle-coupled vibration efect.

Conclusions
In the present study, the authors explored the application of an improved time-domain identifcation method to a structure-vehicle system, which is a simplifed kinetic model of the novel structure AERORail. Te ability of the ITDM is verifed under the moving loads and ¼ vehicle cases. Te main conclusions are as follows: (1) Te acceleration curves demonstrate that the acceleration oscillation of the beam gradually increases as the moving load moves and gradually decreases after the load passes the midspan, and that the dynamic defection increases to its maximum when the moving load moves near the midspan, after which it gradually returns to the equilibrium position and performs free damped vibration. (2) Te simplifed dynamic numerical model of AERORail meets the requirements for carrying out Te above conclusions are based on the estimated structural properties (stifness, modes, and weight) of the AERORail and the ODE-based structure-vehicle model. Although the verifcation is limited to the numerical domain, it still shows the potential of the future application of the ITDM to the real AERORail and other civil engineering structures alike.

Data Availability
Te data used to support the fndings of this study are included in the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.