Vibration Control of the Steel-Concrete Composite Box Girder Bridge with Slip and Shear-Lag Effects by MTMDs under the Train-Bridge Interaction

Steel-concrete composite girder bridges, in essence, are thin-walled structures, the dynamic responses of which will be greater than those of a typical Euler–Bernoulli beam under the train-bridge interaction because of the shear lag and interface slip. Tus, in this study, the dynamic analysis model of a train-composite box girder bridge-multiple tuned mass damper (MTMD) coupling system is proposed, with the derived dynamic equations of this time-varying system. Meanwhile, the program is compiled using the Newmark-β method for the solution, combined with the optimization toolbox to solve the complex optimization design problems of MTMDs involved. Finally, factors afecting the vibration-damping efect are studied, such as the mass ratio of MTMDs, the number of trains, and the slip and shear lag of the composite box girder bridge.

In addition, the steel-concrete composite box girder bridge is one of the thin-walled structures with relatively low stifness. Tus, its special material and section properties also determine its special mechanical behaviors, i.e., shear lag due to a large width-span ratio [14][15][16], and interface slip caused by the fatigue deformation of shear studs between the concrete slab and the steel beam [17][18][19], as illustrated in Figure 1. To a certain extent, its dynamic responses under the train-bridge interaction will be greater than those of a typical Euler-Bernoulli beam because of these efects above [20,21], and more importantly, cannot be ignored. Terefore, more reasonable and reliable steps must be taken to control the vibration of the composite girder bridge.
So, to achieve this, a simple and economical way, at frst, was to install damping devices on the main structure, such as a tuned mass damper (TMD). In the past, Kwon et al. [22], Chen and Huang [23], Shi and Cai [24], Moghaddas et al. [25], Krenk and Høgsberg [26], Chun et al. [27], and Lievens et al. [28] have conducted intensive research on the classical TMD model and its mechanisms, which are relatively mature now. In fact, the TMD is realized by tuning its frequency at or near the corresponding one of a structure (generally the frst-order natural frequency) to control its vibration. Obviously, it is a must to determine the frequency of a bridge accurately during the design stage of the TMD since it is sensitive to the frequency. However, in many engineering cases, due to errors of estimation during construction and the time-varying efect of a coupling system, the TMD may have a pronounced detuning efect, which greatly reduces the vibration-damping efect, and stability and reliability itself are not that strong.
On the contrary, an MTMD system composed of multiple smaller TMDs can be confgured to efectively avoid these problems above because it has a larger bandwidth and aims at more excitation's frequencies than a TMD, not only dispersing its mass and volume but also possessing robustness to suppress the vibration better. Terefore, installing MTMDs on the bridge can validly reduce the resonant responses corresponding to the structure's mode [29,30]. Yau and Yang [31,32] developed a broadband MTMD system to control the vibration of continuous truss bridges under moving loads excited by trains. Lin et al. [29] found that once axle loads of trains are evenly distributed, MTMDs are more efective and reliable than an individual TMD to suppress the resonant responses of the bridge. Li et al. [33] discussed the TMD's dominant factors in an MTMD system and eventually obtained the optimum parameters of MTMDs in controlling the resonant responses of railway simply supported beam bridges induced by highspeed train loads. Luu et al. [34] were devoted to the optimization for high-speed railway bridges to control its multiresonant peaks by an MTMD system. Miguel et al. [35] studied the novel robust design optimization of MTMDs with the principle of maximum entropy and the frefy algorithm, successfully employing it into vehicle-bridge coupled random excitations. Kahya and Araz [36] presented a series of multiple tuned mass dampers (SMTMDs) aimed at high-speed railway continuous bridges and found that the SMTMD system is more capable of controlling the vibration than a typical TMD.
According to the studies mentioned above, the current focus of vibration control is mainly on concrete or steel bridges under train-bridge interactions, and the analysis model is relatively simple. Nevertheless, now, there are few steel-concrete composite girder bridges whose slip and shear-lag efects have not been considered, resulting in an inaccurate analysis of dynamic responses and thus the vibration-damping efect. Given the lack of available methods for vibration control aimed at the steel-concrete composite girder bridges under train-bridge selfexcitations, its mechanisms need to be further studied. However, the composite girder bridge-train-MTMD system is a complex time-varying and coupled system, which renders it more difcult to optimize the parameters, and little else does so.
Hence, considering slip and shear-lag efects, in this study, the dynamic analysis model of a train-composite box girder bridge-MTMD coupling and time-varying system is proposed. Ten, the matrices about the composite box girder bridge are derived to obtain the dynamic equations of this system. Meanwhile, the solution program is developed by employing the Newmark-β method, combined with the optimization toolbox to solve the problem of this complex optimization design problems with MTMDs involved, providing a novel approach for the optimization of the coupled time-varying system. Finally, the optimization method is employed in the numerical simulation to explore the factors afecting the MTMDs' vibration-damping efect based on train-bridge interactions, such as the mass ratio of MTMDs, the number of trains, and the slip and shear lag of the composite girder bridge.

Dynamic Analysis Model of the Composite Box Girder
Bridge. According to the research by Gara et al. [37], the steel-concrete composite girder bridge is located in the Cartesian coordinate system (within three directions), and its geometric parameters are shown in Figure 2. In this coordinate system, the position vector of any point within the bridge can be demonstrated as where ii, jj, and kk are the unit vectors along three directions in Figure 2, respectively, A c and A s are the areas enveloped by the concrete slab and the steel beam, and L is the bridge's span.
To consider the shear-lag efect (see Figure 3), the shear warping intensity functions f c and f s and shear warping shape functions ψ c (y), ψ s (y) are introduced, respectively, according to equations (2) and (3):  Structural Control and Health Monitoring Tus, combined with Figure 4, the translational displacement of any point in the composite girder bridge along three directions are where m � c, s (referring to concrete and steel), u 0 and v 0 are the displacement of the composite box girder bridge along O-y and O-z directions, respectively, y C T and z C T are positions of the torsion center in the transformed section of the bridge along O-y and O-z directions, respectively, y c , y s are positions of centroids along the O-y direction, respectively, z c and z s along the O-z direction, w c0 and w s0 are displacements of centroids along the O-x direction, and ϕ is the angle of the bridge's free torsion (subscript c refers to the concrete slab and s to the steel beam).
Since there is no shear lag at the web of the steel beam, the interface slip displacement Δ sh can be simplifed as where h o is the distance between centroids the concrete slab and the steel beam along the O-z direction.
On top of that, concrete and steel are always linearly elastic. As the shear studs are evenly arranged, its shear connection stifness ρ sh is constant along the O-x direction. Hence, its bond-slip force q sh (x, y, z) is According to the principle of virtual work, the virtual work equation is where σ is the stress tensor, ∇ is the gradient operator, f b and f s are the body forces and surface forces, and V and zV represent its volume and surface area, respectively. δd b represents the variation of the generalized displacements d b , which is  Structural Control and Health Monitoring Ten, taking advantage of the fnite element method and research by Zhu et al. [38] to solve equation (7), the girder bridge is discretized evenly into 3D fnite beam elements (as shown in Figure 5), with two nodes and 18 degrees of freedom (DOFs) including its slip and shear-lag efects, each with 9 DOFs: w cm , w sm , ϕ m , u m , φ m , v m , θ m , f cm , and f sm , where φ m and θ m are rotations around the O-y and O-z axes, respectively; subscripts i and j represent two nodes of the beam element, respectively.
Finally, the stifness matrix K bb and mass matrix M bb can be obtained, respectively. Ten, its damping matrix C bb is derived from Rayleigh viscous damping theory (the derivations and expressions of all matrices and nonzero elements in research by Zhu et al. [38]).

Dynamic Analysis Model of a Train-Composite Box Girder
Bridge Coupling System. Te dynamic analysis model of a fne train-composite box girder bridge coupling system was adopted, as illustrated in Figure 6. As mentioned above, the composite box girder bridge model referred to Section 2.1 and the train were simulated according to a classical 27-DOF vehicle rigid body, including one car body (5 DOFs), two bogies (5 DOFs each), and four wheel sets (3 DOFs each); the symbolic representations of all geometric and characteristic parameters were consistent with the research by Zhu et al. [38]. Terefore, the dynamic equilibrium equations of this coupling system are   where M vv and K vv are matrices about the train's mass and stifness in the global coordinate system, respectively, then, the damping matrix C vv is Rayleigh viscous damping matrix, and q v and q b are displacement vectors of the train and the bridge, respectively. Te dots on their top denote the derivative with respect to time. F v and F b are load vectors of each wheel set and the bridge in the global coordinate system, respectively. We move some components of the load vector in equation (9), so that all elements about dynamic responses to be solved are on the left of equation (9), leaving the track irregularities on the right, and then, equation (9) owns the following form: where the superscript t indicates that nonzero elements within the matrix change as time goes. Te results of its submatrix can be found in the study by Zhu et al. [38].

Consideration of the Rolling Contact.
To better simulate the actual engineering situation, consider the coupling characteristics, and obtain exact responses of the system as trains pass over the composite girder bridge at a high speed, a more accurate wheel-rail rolling contact was introduced. Specifcally, there is no relative displacement between the wheel set and the track both in the foating and rolling directions. However, in the yawing direction, based on Kalker elastic accurate creep theory [39], under the train coordinate system, the relationship between the wheel's lateral relative speed to the rail and the creep force F cy (shown in Figure 7) was assumed to be approximately linear and expressed as where F cy is the creep force in the y direction (yawing direction) under the train coordinate system, F cy is the creep coefcient in the y direction, ξ cy is the creepage, V is the train's operation speed, _ u vwijk is the speed at the kth wheel set of the jth bogie in the ith car body, and _ u b (x vwijk ) is the bridge speed of the bridge corresponding to the kth wheel set of the jth bogie in the ith car body.
Terefore, the independent DOFs of the wheel set are only u vw (yawing direction); v vw (foating direction) and θ vw (rolling direction) can be determined by the displacement of the bridge. In other words, the train model has 19 independent DOFs.

Validity of the Dynamic Analysis Model.
Since this dynamic analysis model has been verifed by an actual case in the research by Zhu et al. [38], where the Italia ETR 500Y train (V � 288 km/h) passed over a steel-concrete composite box viaduct (7 × 46 � 322 m), the dynamic responses in time history agreed with the measured data, proving that the model was close to the real situation with validity.
Despite dealing with a more accurate wheel-rail rolling contact, this study only considered the transverse creep force in the direction, which made little diference in the vertical responses of the bridge. So the proposed model can still be efectively applicable to the following research.

Dynamic Analysis Model of a Train-Composite Box Girder Bridge-MTMD Coupling System with Special Mechanical Behaviors
3.1. MTMD System. Figure 8 illustrates an individual TMD device in the MTMD system installed on the composite box girder bridge. Dynamic equations of its vertical vibration can be given as where m ti , c ti , and k ti are the mass, damping coefcient, and stifness of an individual TMD, respectively, v ti is the absolute vertical displacement of a TMD, and v bl is absolute S te e l B e a m C o n c r e te S la b where m bl , c bl , k bl are the consistent mass, damping coefcient, and stifness of the lth beam element corresponding to where MTMDs are attached, respectively, F bl is the lth beam element's node load vector exerted by wheel sets because of track irregularities, and F T is the vector about the inertial force of MTMDs applied to the beam element, which can be expressed as where δ (−) is the Dirac function given as follows: Substituting equations (15) and (14) into equation (13), it can be simplifed as

A Fine Train-Composite Box Girder Bridge-MTMD Coupling System with Special Mechanical Behaviors.
If a high-speed train operates at a uniform speed V on a composite girder bridge, any position within which the displacement of the track contacted by every wheel set is the superposition of displacements corresponding to bridge and track irregularities. In addition, MTMDs are hung inside to mitigate the bridge's vibration (see Figure 9). In this way, the dynamic equilibrium equation is where F t is the column vector about inertia forces of MTMDs applied to the corresponding beam elements. We move some components of the load vector in equation (17), so that all elements about dynamic responses to be solved are on the left, leaving the known track irregularities on the right. In addition, equation (17) has another following form: where q t is the displacement vector of MTMDs. From equation (18) above, not only are some elements of the composite box girder bridge coupled with those of trains but also coupled with those of MTMDs. Meanwhile, the existing train components and bridge components themselves will be updated with the passage of time. Terefore, it has proved as a time-varying and coupled system.
To solve the second-order complex diferential equations above, the Newmark-β method can be used in the iterative process. Te elements of the stifness submatrix K tt , mass submatrix M tt , and damping submatrix C tt in the MTMD system and the nonzero MTMD-composite girder bridge coupling elements are elaborated in Appendix.

Te Numerical Example with Related Details.
To investigate systematically the infuence of the MTMD's mass ratio, the train's number, and the composite girder bridge's interface slip and shear lag on vibration control, according to the study by Tang et al. [41], an example is taken that a series of German ICE3 trains (all the parameters listed in Table 1) run over a mono-track railway simply supported by a steel-concrete composite box girder bridge (L � 40 m), as shown in Figure 10. Te track irregularities refer to the sixth-grade power spectral density (PSD) according to the U.S. railway standard.
v ti

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An imaginary composite box girder bridge was selected, and its cross-sectional dimensions are illustrated in Figure 10. Te diaphragms were arranged at both ends, and the shear studs were distributed close and spaced evenly. Tus, we assume the shear connection stifness ρ sh (actually the shear force per interface [37]) to be linearly elastic, uniform, and constant. Considering Ollgaard's nonlinear load-slip relationships [42], the load-slip relationship of each shear stud was the secant stifness at 40% of the ultimate strength [43], as ρ sh � 10 kN/mm 2 . Young's modulus of steel and concrete was E s � 2.06 × 10 5 MPa and E c � 3.86 × 10 4 MPa, respectively. In addition, Poisson's ratios are μ s � 0.3 and μ s � 0.2, respectively. Terefore, the frst-order natural frequency obtained from the eigenvalue analysis was 3.98 Hz, with the equivalent composite damping ratio ξ b considered to be 2.98% [44]. Te corresponding vibration mode was vertical bending, selected as the module to be controlled. Te MTMD device was installed on the box girder bridge's interior at its midspan (where the dynamic response is the largest in the frst mode) with a uniformly distributed mass ratio of 0.7%.
As the train operates at a uniform speed V, forming a series of regular axle loads, they might well exert a periodic dynamic efect on the bridge. We assume that the interval between moving train loads d v and its cycle should be d v /V. When it corresponds to the nth-order natural frequency of the bridge or i times of the harmonic cycle, resonance will occur. According to Frýba [45], the critical speed V re (km/h) can be expressed as where f b (Hz) is a certain natural frequency of the bridge (in this study, it is selected as the fundamental frequency) and d v (m) is the fxed wheelbase.
Let i � 2, and the operation speed is V ≈ 180 km/h.

Comparison and Selection of the Optimization Algorithm.
In most cases, closed-form expressions from Den Hartog [46] were utilized in this study to simplify the process of the numerical simulation, appropriate for the optimization of an SDOF (single degree ofreedom) TMD with undamped SDOF structures subjected to harmonic excitations. Tus, the optimum tuning frequency ratio β ti and the optimum damping ratio ξ ti of a TMD are expressed, respectively, where μ ti is the mass ratio of a single TMD, defned as μ ti � m ti /m b (i � 1,2,3, . . . , n), and m ti and m b are the mass of an individual TMD and the composite girder bridge, respectively. Te expressions of Den Hartog or Warburton [47] can be utilized as the initial parameters of optimization in this study. However, Warburton's expressions only apply to SDOF structures subjected to white noise excitations, which is not suitable for this study. Tus, the optimum tuning frequency ω ti,opt , stifness k ti,opt , and damping c ti,opt of a single TMD can be calculated as follows: where ω b1 is the bridge's angular frequency in the frst order, For most of the simple structures, the mode superposition method is appropriate for solving their dynamic responses. Undoubtedly, its dynamic amplifcation factor can be treated as the optimization objective function during the optimization for the TMD. Te advantage of this approach is that only several key modes need tuning for vibration control and that the solution can be obtained through general optimization algorithms. However, in this study, the proposed approach was established for the purpose of higher analysis accuracy, the downside of which was more difcult to optimize the parameters of MTMDs. Obviously, almost all simple optimization algorithms are not a good choice for the optimal design, especially under train-bridge interactions. For another, the time-varying coupling system itself is rather complex in terms of the optimization design. So objective functions can only take the maximum vertical responses after all time steps of iteration, which are nondiferentiable and classifed as implicit function optimization.
For the better optimization and comparison, three trains (n v � 3) passed over the composite box girder bridge, an individual TMD (n � 1) considered a spring-mass-damping system. Terefore, taking the maximum acceleration of the bridge as an objective function, the fnal optimization problem is simplifed as the following expressions:

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At present, numerical optimization methods, such as ergodic search methods, are reasonable for calculations, as the classical closed-form expressions of TMD above are always limited in practice.
In a TMD/MTMD system, the pattern search method (PSM), particle swarm optimization (PSO), and genetic algorithm (GA) were selected as optimization methods separately for programming in MATLAB 2021 embedded with the optimization toolbox. Te specifc implementation is to set the initial TMD's/MTMD's parameters μ t0 , β t0 , ξ t0 , and their boundaries are included within a certain range in equation (21). In the end, optimum TMD/MTMD parameters could be obtained. More details about the procedures are shown in Figure 11.
Under the same initial conditions, three diferent algorithms above were used to optimize μ t , frequency ratio β t1 , and damping ratio ξ t1 of a TMD system. Te optimization results of these algorithms are listed in Table 2, and the optimization results are illustrated in Figure 12.
In light of efciency of optimization, PSM and PSO are better than GA in searching the optimal value, but the optimization time of PSM was signifcantly reduced to the minimum out of these because the initial points were calculated by closed-form expressions of Den Hartog, accelerating the speed of converging to the optimal value points during the search process. Instead, from the optimization results, the computational efciency of GA was still the lowest and was not suitable for the optimization of this strongly coupled time-varying system.
Te goal of the optimization algorithm is to make the results as close to the optimal value as possible with less computation and time. Terefore, the subsequent research in this study adopted PSM combined with closed-form expressions of Den Hartog as the initial point, which saved a great deal of time and facilitated optimization.

Te Mass Ratio.
To lower the difculty of optimization and refect the efect of the mass ratio, the parameters of both trains and the simply supported composite box girder bridge remained consistent with those described in Section 4.1, while the MTMD system was simplifed to a TMD system, i.e., n � 1, installed inside the composite box girder bridge at its midspan. Te range of the mass ratio μ t varied from 0.005 to 0.05 with an interval of 0.005. Te corresponding optimum tuning frequency ratio β t,opt and damping ratio ξ t,opt were obtained by PSM. In this case, the maximum acceleration and defection of the bridge at its midspan were regarded as the optimization objective functions, respectively. Te optimization problem was fnally simplifed to the following expression: min v bl, max μ t , β ti , ξ ti or € v bl, max μ t , β ti , ξ ti , s.t. μ t � 0.005: Δμ t � 0.005: 0.05 ; After parameter optimization by PSM, the maximum dynamic response in the midspan cross section under different mass ratios is shown in Figure 13.
Te results indicate that the vertical dynamic responses of the composite girder bridge at its midspan decrease with an increase in the mass ratio μ t and that the vibrationdamping efect of the TMD system gradually enhanced. However, when the TMD mass ratio reaches 3.5% or more, the trend of acceleration is slightly diferent from defection. Te former is not obvious; the latter, nevertheless, is still decreasing slightly.
From a great deal of earlier research, the mass ratio μ t of the MTMD system usually lies within the range of 1%-5% [48][49][50][51][52]. Te lighter the mass, the more difcult it is for the MTMD system to achieve the ideal damping efect. On the other hand, if the mass is too heavy, like static loading, the dynamic characteristics of the bridge and its mechanical behaviors would change. Afterward, for large structures such as the railway bridge (the mass itself is quite large), it is extremely difcult to obtain a mass ratio of more than 3%. Terefore, the design of MTMDs should be taken into a comprehensive consideration of the economy, bearing capacity of the composite box girder bridge, and vibration-damping efect of MTMDs. Tus, based on the consideration above, in the following research, the mass ratio μ t of the MTMD system will be maintained at approximately 2%.

Train's Number.
Given the limited internal space of the composite box girder bridge, inconvenient installation, and maintenance of an individual TMD, the MTMD system was only composed of three TMDs, which were evenly distributed in the transverse direction. During the optimization process, every small TMD had the same mass ratio of 0.7% and remained constant but not the same frequency β ti and damping ratio ξ ti . Te parameters of both trains and the simply supported composite box girder bridge remained consistent with those in Section 4.1, while the number of trains varied, i.e., n v � 1, 3, 5, and 8, respectively. In this case, the maximum acceleration and defection of the bridge at its midspan were regarded as the optimization objective functions, respectively. Te optimization problem was fnally simplifed to the following expression: After parameter optimization by PSM, the optimum tuning frequency ratio β ti and damping ratio ξ ti of MTMDs and the maximum dynamic response of the bridge at its midspan € v bl, max and v bl, max are demonstrated in Table 3. Te vertical accelerations and defections in both the time history and frequency domain are displayed in Figures 14-21.
Tus, it can be viewed from the above that the interaction between every train and the composite girder bridge will be more obvious due to the increase in the number of trains, and vertical dynamic responses of the bridge are also amplifed with the superposition of continuous periodic excitations from the trains. For example, vertical acceleration and defection were increased to 1.7059 m/s 2 and 5.867 mm, respectively.
However, due to the coupling efect and the assumption of the wheel-rail rolling contact, the bridge's natural frequency is always changing when the train runs on it, because the mass of wheel sets is added to the bridge so that the dynamic peak in the frequency domain does not correspond to the frst-order frequency of the bridge but is slightly smaller. Even so, MTMDs still work as they would frst resonate with the excitation to suppress the vibration of the corresponding mode of the composite girder bridge and avoid resonance after tuning. Figure 22 shows vertical defection and acceleration damping ratios of the composite box girder bridge at its midspan regarding diferent train's numbers within the whole time trains pass. When the number is increased from 1 to 3, the vibration damping ratio for the vertical acceleration of the bridge at its midspan reduces from 34.39% to 33.52% and for the vertical defection from 1.62% to 1.51%. Although they are both reduced, yet not much, they tend to be toward stability. When the number is 3, the vibration damping ratios of its vertical acceleration and defection are close to the minimum. When the number increases from 3 to 5, the vibration damping ratio for the vertical acceleration increases to 40.98% and for the vertical defection to 14.36%. When the number of trains is 8, the vibration damping ratios for vertical acceleration and defection become maximum, which are 43.02% and 16.53%, respectively. In general, the vibration damping ratios of vertical acceleration and defection of the bridge at its midspan increased as the number of trains increased. When the number was increased from 1 to 8, the resonance efect became more obvious. Tis made the vibration damping ratios of MTMDs for the vertical acceleration and defection increase by 8.63% and 14.91%, respectively, whereas the dynamic characteristics of the composite girder bridge do not change in nature.
In addition, the vibration-damping efect of MTMDs for vertical acceleration is still better than that of vertical defection, because MTMDs themselves only reduce the dynamic defection of a structure, and static defection accounts for a large proportion. In other words, the response of vertical defection was particularly concentrated in the range of low frequency (0-1 Hz), but MTMDs can be   brought into play only when their frequencies fall in or near a certain frequency of the bridge, so its vertical defection would not be reduced so much.

Interface Slip.
As illustrated above, the MTMD system was still composed of three TMDs, which were evenly distributed in the transverse direction. During the  Figure 13: Infuence of the mass ratio.  Structural Control and Health Monitoring optimization process, every small TMD had the same mass ratio of 0.7% and remained the same but not the frequency β ti and damping ratio ξ ti . Te parameters of trains remained consistent with those in Section 4.1, while the shear connection stifness of the bridge varied, i.e., ρ sh � 1 kN/m 2 , ρ sh � 5 kN/m 2 , ρ sh � 10 kN/m 2 , and ρ sh � 100 kN/m 2 .
In this case, the maximum acceleration and defection of the bridge at its midspan were considered the objective functions for optimization. After parameter optimization by PSM, the optimum tuning frequency ratio β ti and damping ratio ξ ti of MTMDs and the maximum dynamic responses of the bridge at its midspan € v bl, max and v bl, max are demonstrated in Table 4, respectively. Te vertical accelerations and defections in both the time history and frequency domains are displayed in Figures 23-30.
Tus, it is clear from above that the vertical defection of the bridge at its midspan decreases originally and then increases slightly with the shear connection stifness ρ sh getting larger, as minimum 5.862 mm and maximum 6.071 mm; the larger the interface connection stifness, the slighter the increase. However, vertical acceleration increases with an increase in ρ sh , as the maximum is 1.7179 m/ s 2 ; the larger the interface connection stifness, the slower the increase. Te most likely reason is that when a weak shear connection (ρ sh � 1 kN/m 2 ) changes to a strong shear connection (ρ sh � 100 kN/m 2 ), its fundamental frequency increases slightly, and V re gradually approaches V, so the dynamic responses are amplifed, making it close to the condition of resonance. Although vertical acceleration increases to a certain extent as well as vertical defection, resonance is not prominent due to the large proportion of static (or low-frequency) components. So the overall trend has no signifcant diference. Figure 31 shows, respectively, vertical defection and acceleration damping ratios of the composite box girder bridge at its midspan regarding diferent shear connection stifnesses within the whole time trains pass. When ρ sh is 1 kN/m 2 , the operation speed of trains V is far from the critical speed V re , the MTMD system has a detuning efect, and the vibration damping ratios decrease to minimum, which are 18.96% and 2.75%, respectively. When ρ sh increases from 1 kN/m 2 to 5 kN/m 2 , the damping ratio for the vertical acceleration of the bridge at its midspan increases to 39.69%, and for vertical defection, it increases to 14.50%, revealing an obvious diference. When ρ sh increases from 10 kN/m 2 to 100 kN/m 2 , the vibration damping ratio tends     Structural Control and Health Monitoring to be stable. When ρ sh is 100 kN/m 2 , V is the closest to V re , and the resonance efect becomes the most obvious, so the vibration damping ratios of vertical acceleration and defection reach maximum 45.95% and 18.72%, respectively.
In general, the interface slip reduces the vibration damping ratios of MTMDs for both vertical acceleration and defection by 26.99% and 15.97%, respectively. Te vibration damping ratios of vertical acceleration and defection of the bridge at its midspan increase as ρ sh becomes larger. In addition, the larger the shear connection stifness, the slighter the increase. Terefore, the decrease of ρ sh slightly changes the dynamic characteristics of the bridge, making its resonance weaker and even less obvious. Tus, the vibration damping ratio of MTMDs greatly reduces, which, to a certain extent, refects that it is only suitable for structures under narrow bandwidth excitations.

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Structural Control and Health Monitoring 5.4. Shear Lag. As illustrated above, the parameters of trains remained consistent with those in Section 4.1, and the shear connection stifness ρ sh is 10 kN/m 2 , while width-span ratios of the bridge varied, i.e., 2b c /L � 0.25 and 2b c /L � 0.5. In this case, the maximum acceleration and defection of the bridge at its midspan were considered the optimization objective functions. After the optimization by PSM, Tables 5 and 6 specifcally show the optimum tuning frequency ratio β ti and damping ratio ξ ti of MTMDs and the maximum dynamic responses of the bridge at its midspan € v bl, max and v bl, max , comparing the results with a shear-lag efect with those without. Te vertical acceleration and defection in both the time history and frequency domains are displayed in Figures 32-39.
Tus, it is clear from above that there is some diference between these two conditions. When 2b c /L � 0.25, shear lag of the bridge appears not as pronounced, so the vertical maximum acceleration of the bridge at its midspan increases slightly, and the vertical maximum defection decreases from 6.109 mm to 6.086 mm, which has little diference because shear lag does not reduce the stifness of the bridge too much. Tus, the vibration damping ratio of MTMDs for vertical acceleration decreases from 35.12% to 33.39% and that for vertical defection decreases from 15.65% to 15.63%. However, the MTMD system still plays a role in vibration control. When 2b c /L � 0.5, shear lag appears more serious, and the stifness of the bridge decreases more, which made the vertical maximum acceleration of the bridge at its midspan increase from 0.9625 m/s 2 to 1.0167 m/s 2 , and the vertical maximum defection from 2.806 mm to 3.189 mm but not drastically. However, because shear lag greatly changes the dynamic characteristics of the bridge, the components in the vertical defection of the bridge increase, and even the peak does not correspond to the fundamental Structural Control and Health Monitoring frequency. To be specifc, the vibration damping ratio of MTMDs for vertical acceleration decreases from 35.66% to 24.26% and that for vertical defection decreases from 16.91% to 6.37%, suggesting that the detuning efect is pronounced and even MTMDs could not control the vibration of the bridge, especially for vertical defection. Figure 40 shows a comparison of the vibration damping ratios in the time history between the bridges with and without shear lag. For the vertical acceleration of the bridge at its mid-span, the larger the width-span ratio 2b c /L, the greater impact shear lag has on the vibration-damping efect. When the width-span ratio 2b c /L changes from 0.25 to 0.5, the vibration damping ratio considering shear lag of the bridge simply decreases from 33.39% to 24.26%. However, the results are worse for vertical defection, which drops rapidly from 15.63% to 6.37%.
Terefore, as 2b c /L increases, shear lag afects the distribution of components of the dynamic responses in the frequency domain. Tis would be aggravated with an increase in 2b c /L, and the vibration damping ratio for vertical defection is seriously weakened. Moreover, in practice, shear lag of the composite girder bridge should be fully taken into consideration of vibration control using MTMDs.

Conclusions
In this study, the dynamic analysis model of a complex traincomposite box girder bridge-MTMD coupling system is proposed with the derived dynamic equations for the following research. Trough the numerical simulation, conclusions can be drawn as follows: (1) In terms of the strongly coupled and time-varying system, PSM and PSO are better than GA in searching for the optimum parameters to suppress the vibration caused by train-bridge interaction. But the optimization time of PSM is reduced to the minimum, taking advantage of the closed-form Den Hartog's expressions. (2) Te vibration damping ratio for vertical dynamic responses of the bridge increases as the train's number is larger since resonance becomes more obvious, whereas the dynamic characteristics of the bridge do not change in nature. (3) Te vibration damping ratio for vertical dynamic responses of the bridge increases as ρ sh is larger. Meanwhile, the larger the shear connection stifness, the smaller the increase. However, the decreased shear connection stifness slightly changes the dynamic characteristics of the bridge and reduces the vibration-damping efect of MTMDs. (4) When 2b c /L is small, shear lag of the bridge is not pronounced. So the vibration-damping efect of MTMDs slightly decreases, but the device still plays a role. When 2b c /L is large, shear lag of the bridge becomes obvious, decreases the bridge's stifness, and greatly changes its dynamic characteristics, even the peak of the dynamic response in the frequency domain does not correspond to the fundamental frequency. (5) Under the train-bridge interaction, both slip and shear lag have a signifcant infuence on the vibration-damping efect of MTMDs. Hence, more attention should be given during every structural stage, such as design, construction, and operation to minimize its hazards.

Appendix
Te results of the submatrices of stifness K tt , mass M tt , and damping C tt of the MTMD system are as follows:   where n is the total number of TMDs.
Te nonzero elements of the stifness submatrix and damping submatrix in the composite box girder bridge-MTMD coupling system are as follows: for i � 1: n K bt (i, n bl ) � K bt (i, n bl )-k t (i); K t bb (n bl , n bl ) � K t bb (n bl , n bl ) + k t (i); C bt (i, n bl ) � C bt (i, n bl )-c t (i); C t bb (n bl , n bl ) � C t bb (n bl , n bl ) + c t (i); end K tb � K T bt C tb � C T bt where k t (i) � k ti , c t (i) � c ti , n bl is the n bl -th degree of freedom of the beam element of the composite box girder bridge where the ith TMD is installed, superscript T represents the transpose of a matrix, and subscript bt represents the elements of the composite box girder coupled with installed MTMDs.

Data Availability
Te data supporting the current study are available from the corresponding author upon reasonable request.

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