The Parameter Identification of Structure with TMD considering Seismic Soil-Structure Interaction

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Introduction
Structural control technologies are commonly adopted to mitigate the induced vibrations of structures caused by earthquake, and many novel devices for structural vibration control have been developed in the last few decades [1][2][3][4][5].Te tuned mass damper (TMD) consisting of mass, stifness, and damping components is one of the most promising and efective devices, and its vibration control performance has been widely investigated [6].However, the variation in primary structure and TMD properties would greatly infuence TMD's performance on the suppression of structural vibration [7][8][9].
Parameter identifcation is crucial to estimate the dynamic characteristics of a structural system through ambient vibration measurements, which can serve as a basis for condition assessment, structural damage detection, and long-term health monitoring.Many advanced methods have been utilized for the parameter estimation of civil engineering structures [10][11][12][13][14]. Love et al. [15] estimated the inherent structural damping of structure-TMD system using the random decrement technique.Weber et al. [16] assessed the long-term performance of pedestrian bridge with TMDs by identifying system parameters.Wang et al. [17] utilized a Bayesian method to obtain the modal parameters of the primary structure with TMD.Yuan et al. [18] integrated the second-order blind identifcation method with the empirical wavelet transform to get the modal frequencies and damping of structure with TMD.Cao et al. [19] used the stochastic subspace technique to identify the modal properties of the coupled structure-TMD system.Cho et al. [20] derived decoupled equations of motion for identifying dynamic properties of secondary mass dampers by the full-scale feld test.Te methods proposed in the above studies for identifying the parameters of the structure-TMD system have good identifcation accuracy in specifc cases.However, the iterative process of most methods is complicated, and some methods show low accuracy in identifying parameters at high frequency range.Rofel et al. [21] obtained the modal properties of structures with a pendulum TMD using the Extended Kalman Filter.Schleiter et al. [22] employed an adaptive unscented Kalman flter scheme to identify the system parameters for variable stifness TMDs.Hwang et al. [23] extracted modes using a modal-based Kalman flter for damped structures.As one of the most commonly used identifcation methods, the fltering methods are computationally intensive and not suitable for perform multi-degree-of-freedom identifcation.
Te soil-structure interaction (SSI) infuences the dynamic characteristics of the structure [24], such as period and damping, and seismic responses of the structure [25].In the soil-structure-TMD system, the seismic response of the structure is afected by soil, TMD, and their coupling efects.Consequently, it is important to consider the SSI efects on structure equipped with TMD, especially on fexible soil.Liu et al. [26,27] and Jabary et al. [28,29] conducted shaking table tests and geotechnical centrifuge tests to explore the efectiveness of TMD for multi-story frames considering SSI efects, respectively.Jia et al. [30] discussed the infuence of a variety of parameters on TMD performance including SSI efects by a fully 3D model.Abd-Elhamed et al. [31] compared the seismic response of TMD controlled building with that of uncontrolled case.Gorini et al. [32] established a general nondimensional formulation for the linear soilstructure-TMD system.Khoshnoudian et al. [33] analyzed the diferences of seismic responses of building structures under three foundation conditions.Zhang et al. [34] carried out a nonlinear seismic fragility assessment of a benchmark structure with TMD in the SSI system.Espinoza et al. [35] studied the torsional control performance of TMD involving seismic soil-structure interaction.Bekdas ¸et al. [36] and Djedoui et al. [37] adopted metaheuristic algorithms for the optimization of TMD parameters considering SSI efects.Gao et al. [38] simulated the SSI system employing lumped parameter models to investigate the TMD performance.From the above studies, it can be concluded that SSI efects may become crucial in the seismic response of structure and the seismic control performance of TMD, so including SSI efects would refect the work condition of structure and TMD under earthquake excitations more actually.
Nevertheless, the previous studies on parameter identifcation of structure equipped with TMD have not taken soil-structure interaction into consideration, and the parameter identifcation of structure equipped with TMD in the presence of seismic SSI is rarely reported.Hence, this paper conducted the parameter identifcation of soil-structure-TMD system.Te accelerated particle swarm optimization (APSO) algorithm combined with the search space reduction (SSR) method is proposed for the parameter estimation of soil-structure-TMD system.Furthermore, the frequency response function and transmissibility function are used for output-input and output-only cases, respectively.Te rationality of the identifcation method is verifed by numerical simulation, and some factors on identifcation accuracy are considered.Tis study is of great practical signifcance to develop a parameter identifcation framework of structure with TMD including SSI efects.
Te rest of this paper mainly consists of the following.Section 2 introduces the identifcation methodology in detail.Section 3 describes a brief overview of the numerical model for this study.Te identifcation results are elaborated and discussed in Sections 4 and 5. Section 6 summarizes the conclusions.

Methodology
2.1.Transmissibility Function.Te motion equation of structure with N degrees of freedom (DOFs) can be expressed as where [M], [C], and [K] are the mass, damping, and stifness matrices, respectively, and f(t) is the external force.By Fourier transform, the frequency response function H(ω) can be obtained through the ratio of the measured forces F(ω) and vibration responses X(ω).
Assuming that x i (t) and x o (t) are the response records of measuring points i and o, the transmissibilities are defned as the ratio of the two response spectra [39], namely: where X i (ω) and X o (ω) are response spectra of x i (t) and x o (t), respectively.When multiple excitations are applied on the system, the transmissibility can be expressed as [40] T where H i,m (ω) is the frequency response function corresponding to the input-output transitive relation of the i and m DOFs; F m (ω) represents the Fourier transform coefcients of external excitation at m DOF; and N F is the number of measured points.In this paper, the frequency response function and transmissibility function are used for identifying structural physical parameters for a priori known seismic input (output-input) and a priori unknown seismic input (output-only) situations, respectively.

APSO-SSR.
Te accelerated particle swarm optimization (APSO) algorithm is an improved version of PSO algorithm that avoids the problem of premature convergence, and it has the advantages of concise concept, convenient implementation, and fast convergence speed [41,42].Te main steps of APSO algorithm are as follows [43]: Step 1. Initialize the particle swarm.Set the maximum evolution algebra T max and the population size n.In this study, the population size n and iteration times T max are taken as 100 and 50, respectively.Randomly generate n particle groups X 1 , X 2 , . . ., X n in the defned space R n to form the initial population X(t) (t � 1) and initial velocities V 1 , V 2 , . . ., V n for each particle.

Structural Control and Health Monitoring
Step 2. Evaluate particle swarm X(t).Calculate the ftness value of each particle in the population and compare the current ftness value of the particle with p besti .If it is better, set its position as the current optimal position of particle i in the D-dimensional space.
Step 3.For each particle, compare its current ftness value with the global optimal solution g best of the population.If it is better, update its position to the current global best position of the population.Update the velocity and position of particles to generate a new population X (t+1) .
Step 4. Check if the end condition is met.If it is, end the algorithm and output g best ; otherwise, t � t + 1, go to step 2. Te ending condition is generally that the maximum evolutionary algebra T max or the improvement degree of g best less than the given accuracy ε is achieved through iterative optimization.
Te search space reduction (SSR) method can improve the efciency and accuracy of identifcation algorithms by reducing large parameter search space [44,45].Tus, this study adopts SSR method within the APSO algorithm.A number of APSO independent runs with the same original search space are parallelly conducted to evaluate the values of the objective function and the solutions.A set of solutions is obtained, and the solution with the worst ftness value is removed.Te remaining PR (r) solutions are marked as Φ.
Te weighting coefcient of each solution is expressed as where fit(•) is the ftness value to measure identifcation performance.Te better identifed solution has higher weights.Te weighted mean value and weighted standard deviation of the i th identifed parameter are computed so as to determine the new parameter bounds.
Subsequently, the lower and upper limits of the new search space of the i th identifed parameter are generated: where η denotes the window width coefcient of the new search space, which is 5 in this study.In addition, to avoid that the new search space sometimes exceeds the original boundary, the fnal new space is generated by means of the intersection of the original search space and the new trial search space.
In the fnal new space, the APSO would conduct a new round of parameter identifcation and update the optimal parameter solution.Te parameter identifcation fowchart is displayed in Figure 1.

Mechanical Model.
An N-story superstructure with TMD attached to the top foor considering SSI efects is shown in Figure 2. Te dynamical SSI is complex, and thus the simplifed swaying-rocking substructure model is commonly used in the parameter identifcation of the SSI model [46,47].Te DOFs of primary structure, TMD, and translation-rotation foundation are N, 1, and 2, respectively, so the simplifed mechanical model of soil-structure-TMD system has a total of N + 3 DOFs.
Te motion equation of the N-story primary structure equipped with TMD on the top considering SSI efects can be written as follows: )  4 Structural Control and Health Monitoring A fve-story structure is taken as an example [48].Te story mass m 1 and foundation mass m 0 are taken as 3 × 10 5 kg.Te story stifness from the 1 st to 5 th story is 7k, 5k, 3k, 2k, and k (k � 5 × 10 7 N/m), respectively.Te story mass moment of inertia I i and foundation mass moment of inertia I 0 are 7.5 × 10 6 kg•m 2 .Te stifness damping coefcient is taken as 0.02 without considering mass damping.Te mass ratio, frequency, and damping ratio of TMD are defned as follows: Structural Control and Health Monitoring where μ, f T , and ξ are the mass ratio, frequency, and damping ratio of TMD, respectively; m T , k T , c T , and ω T are the mass, stifness, damping, and circular frequency of TMD, respectively; and M is the mass of structure.
If the TMD mass ratio μ and structural damping ratio β are known, the optimal frequency ratio and damping ratio of TMD can be expressed as follows [49]: where α opt and ξ opt represent the optimal frequency ratio and optimal damping ratio, respectively, and f T and f s are the frequencies of TMD and the primary structure, respectively.In this study, the mass ratio, frequency, and damping ratio of TMD are taken as 0.02, 0.678 Hz, and 21.53%, respectively.Te SSI efects under soft soil condition are more signifcant than other soil types [50].Terefore, soft soil is taken in the structural model.Te formulas for the stifness and damping of foundation are as follows [51]: where k and c represent stifness and damping, respectively; subscripts r and s represent translation and rotation, respectively; v s , V s , G s , and ρ s are Poisson's ratio, shear wave velocity, shear modulus, and density of soil, respectively; and R 0 is the base radius of foundation.Te specifc values of the foundation parameters are shown in Table 1.
As one of the most commonly used methods for simulating nonlinear SSI efects, the equivalent linear methods have been proven to be practical and feasible by many studies [52][53][54][55][56]. Te identifcation method in this study is also applicable for the equivalent linear model.

Nonstationary Earthquake Motion.
Te evolutionary power spectrum model of nonstationary earthquake acceleration process is adopted, and its bilateral evolutionary power spectral density function is as follows: where S U g (t, ω) is the bilateral evolution power spectral density function of the nonstationary earthquake acceleration process and A(t) is the intensity modulation function.
where c and d are the time when the peak acceleration of seismic motion occurs and the index of the control shape, respectively; T represents the total duration of the nonstationary earthquake acceleration process; and ω 0 , ξ 0 , a, b, c, and d can be determined from the site classifcations and design earthquake efects specifed in current seismic code.Te spectral parameters S 0 (t) refecting the intensity of seismic motion can be determined as where a max is the mean value of the peak acceleration of random seismic motion and c is the peak factor.Te parameter values of the evolutionary power spectrum model are listed in Table 2.

Optimization Objective.
Te residual error is introduced to evaluate the frequency response function and transmissibility function between measurements and simulations.Te objective function J(z) is built: where erf represents the error function;  X(θ) is the frequency response function of identifed model under known excitation condition, or its transmissibility function for excitation unknown case; X is the value of the frequency response function or transmissibility function obtained from the actual measurements; and θ represents the physical parameters of the predicted model.Te objective function is 6 Structural Control and Health Monitoring to minimize the frequency response function and transmissibility function between the measured structural response and the simulated response of the identifed mathematical model.

Identification Results
To verify the efectiveness of the identifcation method based on transmissibility functions under unknown excitation Te physical parameter identifcation results of soilstructure-TMD system are shown in Table 3. Te results show that there is no deviation between the identifed and actual values of each parameter for output-input and output-only cases.Tis demonstrates that the APSO-SSR method has excellent applicability and efectiveness in the two situations.Next, the diferences between the two cases (output-input and output-only) in terms of frequency response function and transmissibility function are studied.
Te frequency response function and transmissibility function of simulated and identifed values are compared in Figures 3 and 4. In these fgures, H and T represent the frequency response function and transmissibility function, respectively; corner markers 1, 2, 3, 4, 5, 6, 7, and 8 represent TMD, 1st story, 2nd story, 3rd story, 4th story, 5th story, and the translation and rotation of foundation, respectively; the frst and second corners are the excitation reference point and the response reference point in the frequency response function, respectively, while both corner markers are response reference points in the transmissibility function.In general, the identifed parameters based on frequency response function or transmissibility function are well matched with the numerical simulation values.Tis observation implies that the identifcation methods are successful in obtaining the physical parameters of the soil-structure-TMD system.In this sense, under unknown excitation condition, the parameter identifcation method based on transmissibility function is feasible and can accurately identify the actual system parameters.

Frequency Band.
Te impacts of frequency band on the identifcation accuracy are further considered in this subsection, and the identifcation performance for output-only case at diferent frequency bands is analyzed through comparison with output-input case.Te frst fve frequencies of structure with TMD considering SSI efects are 0.85 Hz, 1.16 Hz, 2.49 Hz, 3.89 Hz, and 5.43 Hz, respectively.Te selected frequency bandwidths are shown in Table 4. Te frequency bands are divided into 0-5 Hz, 5−10 Hz, 10−15 Hz, and 15−20 Hz, respectively.Te structural responses are transformed from the time domain to the frequency domain.Te parameter identifcation is carried out in the frequency domain at the corresponding frequency bandwidth.
Te comparisons of the normalized values (identifed value/actual value) for system physical parameters at different frequency bands between output-input and outputonly cases are displayed in Figures 5 and 6.From these fgures, it can be observed that the identifcation performance at bandwidth I-1 and I-5 is fne for both two situations, while there are varying degrees of errors in I-2, I-3, and I-4 cases, especially at bandwidths I-3 and I-4, as the range does not cover the frst fve natural frequencies of the system.Under unknown excitation condition, the identifcation results are accurate in I-1 case but there are some errors in the identifcation results of the translational stifness, the frequency, and damping ratio of TMD at high frequency bandwidth.At the same time, for input-output case, the identifcation results are also accurate in I-1 case       Structural Control and Health Monitoring but some identifcation errors of ffth story stifness, the frequency, and damping ratio of TMD appear at high frequency bandwidth.It is proved that using high frequency data would cause errors in the estimation results.

White Noise.
Te measured signals are often disturbed by noise in practical engineering.Te infuence of noise on the identifcation performance is explored in this subsection.Te noise is added to the frequency function obtained from frequency domain analysis.
x ′ � x ×(1 + εη), (31) where x and x ′ are the calculated value of mathematical model and the value after adding random noise; ε is the noise level; and η is a normal random variable with zero mean and unit standard deviation.Te white noise cases corresponding to ε are listed in Table 5. Te frequency range of 0-5 Hz is used for analyzing the noise resistance of system parameter identifcation.
As seen clearly from Figure 7, the normalized values of all parameters are almost equal to 1 under II-1∼II-5 noise levels, indicating excellent identifcation performance and noise resistance.Meanwhile, it also can be found that the efects of white noise on identifcation accuracy of  Structural Control and Health Monitoring output-only case are more pronounced than those of outputinput case.Te identifcation result of foundation stifness and damping under high noise level is not satisfactory.Hence, it ought to be emphasized that the identifcation accuracy of foundation parameters using measured signals for output-only case needs to be paid attention to.

Convergence.
To study the convergence of APSO-SSR method and the impact of signal disturbance on parameter identifcation, this subsection takes three noise levels, namely, II-2, II-4, and II-5 cases, to identify physical parameters of the soil-structure-TMD system.In the identifcation process, the convergence criterion of this method is that the search space for structural parameters tends to stabilize.Te parameter variation of II-2, II-4, and II-5 cases during the iterative process is illustrated in Figure 8.It can be seen that the parameter search space shrinks after every 50 iterations.Te search space is continuously updated and reduced through the SSR strategy, gradually approaching the       Structural Control and Health Monitoring true value.Te convergence results of most parameters tend to be stable after 150-200 iterations, which indicates that the parameter search domain achieves excellent convergence with small number of iterations.In addition, under diferent noise pollution situations, the identifed parameters fuctuate greatly at the initial stage of each newly generated search space and gradually converge afterward.

Conclusion
Tis paper proposed an identifcation method for system physical parameters of structure equipped with TMD considering SSI efects based on the APSO-SSR method using frequency response function and transmissibility function.Te physical parameter estimation framework proposed in this study is verifed by numerical simulations.Te following major conclusions can be made: (1) Te proposed identifcation method using frequency response function and transmissibility function provides reliable estimations for the system physical parameters of structure equipped with TMD considering SSI efects for output-input and input-only cases, respectively.(2) Both for output-input and output-only cases, identifcation errors are prone to arise if frequency band range does not cover the frst fve system frequencies.Hence, the frequency band for identifcation should include primary frequencies of the system.(3) Te parameter identifcation strategies have excellent noise resistance as a whole.But white noise interference yields inaccuracy in identifying the foundation stifness and damping, especially under unknown excitation condition.
(4) Te search space reduction method efectively decreases and updates the search space during the iterative process.Although the noise pollution causes instability at the initial stage of newly generated search space, the SSR method improves the convergence speed on the premise of ensuring accuracy.

Figure 2 :
Figure 2: Te mechanical model of the soil-structure-TMD system.

Table 1 :
Foundation parameters.Translational stifness k s (N/m) Rotational stifness k r (N•m) Translational damping c s (N•s/m) Rotational damping c r (N•s•m)

Table 2 :
Parameters of the evolutionary power spectrum model.

Table 3 :
Identifcation results of system physical parameters.

Table 5 :
Te white noise cases.