Performance‐based seismic design of tuned inerter dampers

This paper proposes a novel fully probabilistic framework for the performance‐based seismic design of structures and uses tuned inerter dampers (TID) installed in civil engineering structures subjected to seismic loads to illustrate its applicability. The framework proposed is based on stochastic reduced‐order models, which makes it computationally efficient and can be used for the design of TIDs installed in any complex nonlinear structures subjected to general nonstationary, non‐Gaussian stochastic processes. In this study, the TID is installed in a multi‐degree‐of‐freedom nonlinear structure that is subjected to synthetic seismic records. Numerical results show that the framework proposed is able to provide rigorous and robust values for the parameters of the TID. The design parameters obtained using the stochastic framework proposed are compared with benchmark deterministic approaches, tested also for a large data set of ground‐motion real records. It is shown that the stochastic approach provides insightful designs of the TID that are consistent with the site seismicity and the frequency content of the stochastic excitation.

In addition to the earlier studies demonstrating how TIDs can be designed to improve the behaviour of host structures, such as multistorey buildings 8 or stay cables, 13 TIDs have also been tested experimentally using dynamic substructuring 14 emphasising their feasibility and applicability. Besides civil engineering design applications, 12,15,16 inerter-based vibration absorbers have been considered in the context of a range of practical applications, such as vehicle suspensions, [17][18][19] train suspensions, 20,21 or aircraft landing gears. 22 The majority of the studies on optimal control devices of the kind presented herein consider frequency or time-domain analyses that involve the calculation of multiple structural system's response.
The aim of this study is to propose a novel, general, accurate, yet efficient, fully probabilistic framework for the performance-based design of vibration control systems, such as the ones enumerated above, for civil engineering structures subjected to seismic loads. Once presented, the proposed methodology is then demonstrated using the example of the probabilistic seismic design of a TID. Even though the framework is discussed through the use of this example, it allows for the structural system, the vibration control device, and the input to be replaced by customised models relevant to other applications. Nevertheless, the example shown uses a rather general case of a TID installed in a nonlinear MDOF structure subjected to a non-Gaussian, nonstationary stochastic process.
The seismic performance design of structures is a challenging problem for the following reasons: The number of available earthquake records is insufficient even in high-seismicity areas, 23 and performing dynamic analyses for realistic complex structures is strenuous. These issues have been discussed extensively in the literature, and approximate methods were proposed to overcome them. It is common to use real ground-motion records from large data sets selected to be consistent with specified spectral values of the response. [24][25][26] The limitations regarding the small number of records available are often resolved by methods involving scaling recorded earthquake time histories to common intensity measures. One popular approach, known as the incremental dynamic analysis (IDA) and presented by Vamvatsikos et al., 27 involves repeated scaling of seismic ground motions to increasing intensity measures until the specified damage state is reached. This method may yield unsatisfactory results for some nonlinear systems. 28,29 The limitations of scaling ground motions 30 may be resolved by simulating synthetic ground motions, an approach that has already been adopted in the performance-based seismic analysis of structures. 29,31 The issue of calculating the response of dynamic systems in the general nonlinear systems subjected to nonstationary, non-Gaussian input, because the framework of the random-vibration theory can not be used in complex realistic applications. The general random-vibration theory is insufficient to perform this task, 32 because it can only provide solutions for second-moment properties of the states of arbitrary linear systems subjected to Gaussian random input. Most applications of inerter-based systems were studied in a deterministic framework, where closed form solutions can be found for simplified structures and excitation patterns, 8,[33][34][35] or numerically, 13,14 for more complex scenarios. Some studies used elements of random-vibration theory to design inerter-based vibration-control systems, 36-38 but by adopting this approach, they were limited to linear structures subjected to Gaussian stationary excitations. In a similar context, Masri et al. 39,40 develop relations for the covariance of the response of linear SDOF or MDOF subjected to amplitude-modulated nonstationary Gaussian white noise. Other studies, among which, Wang and Su 41 study nonlinearities of inerters in a similar context by using approximate linear transfer functions. These are just a few examples of analytical relations developed for simplified scenarios of linear or linearised systems subjected to Gaussian white or non-white noise. An early comparison of benchmark methods for the characterisation of the dynamic response of nonlinear MDOF structures is presented by Bazzurro et al. 42 More solutions for dealing with the computational complexity of resolving structural systems were proposed by assuming that the structures behaved linearly, 43,44 by linearization of nonlinear structural systems, [45][46][47] or by using a reduced number of modes of vibration in the response analyses. 48,49 Monte Carlo (MC) is the only general numerical method available for solving complex nonlinear dynamic systems subjected to complex stochastic input and is widely used in many applications. However, the MC solution is computationally expensive, arguably prohibitive so, when applied to the design of complex realistic nonlinear systems 50,51 because it requires repeated deterministic dynamic analyses for many random ground-motion samples. The development of efficient alternatives to MC method has been a topic that has drawn a lot of attention over the years, with various applications, among which a few examples are the quasi MC simulation, 50,52 Latin hypercube, 53 the change of measure 54 (Section 5.4), the equivalent stochastic linearisations, 55 the stochastic averaging method, 56 or reduced orders of the nonlinear system by using linear or nonlinear normal modes. 57,58 The framework proposed in this paper for the seismic performance-based design of the TID relies on stochastic reduced order models (SROMs). 59 SROMs are stochastic processes defined by a relatively small number of unequally-likely samples that fully characterize the probability law of that process. The samples of the SROM are selected such that they reduce the differences between the SROM and the original input process. Thus, only the small number of optimally selected SROM samples are used for the dynamic analyses -this significantly reduces the number of analyses required by the MC method. The method was originally developed for dynamic response, 60 and recently, it was shown that it can be extended 61 and through further refinements can now be used to solve stochastic differential equations. 62 Here, the proposed framework uses an SROM for the seismic input, enabling response statistics to be used as design criteria. The metric used for the design of the TID is defined with respect to the tail distribution of the engineering demand parameters. The sensitivity between the MC and the SROM response statistics is also analysed in order to establish the accuracy and robustness of the method proposed.
The current paper is structured in three main parts. In the first part, the performance-based seismic design framework is described with respect to the structural system chosen: a TID installed in a MDOF nonlinear system. The performance criterion for the design of the TID is to reduce the area under the tail distribution of the maximum displacement of the structural system beyond a critical displacement value, or in other words, to minimise the probability of exceeding a critical maximum displacement. The second part of the paper consists of a description of the seismic hazard to which the structure is subjected. Synthetic ground-motion records are used for the dynamic analyses. These records are samples of nonstationary, non-Gaussian stochastic processes, defined as a function of moment magnitudes, and source-to-site distances. The construction of the SROM for the earthquake excitation model is also described in this section. Finally, in the last part of the paper, numerical results that examine the performance of the TID as the structure is exposed to the SROM excitation samples are shown, and a methodology of selecting the design parameters of the TID is shown in detail. The paper concludes with a discussion of the methodology proposed with respect to benchmark studies and current practices.

PERFORMANCE-BASED SEISMIC DESIGN
The current paper proposes a general framework for the performance-based seismic design of TIDs, installed in structures subjected to earthquake loads. It is important to note that the vibration-suppression performance-based design problem addressed herein is general and can be applied to any type of stable nonlinear structures and stochastic loads, characterised by their probability law or samples. Here, the design of the TID is performed for a general scenario of a nonlinear MDOF system subjected to nonstationary, non-Gaussian seismic input to demonstrate the proposed approach.
In the context of the performance-based framework, the design parameters of the TID, as defined later in this section, will be selected based on the overall performance of the structural system, in terms of the system's response statistics. Let us denote the response of the structural system altered by a TID with parameters D and subjected to the stochastic process A(t), by X(t; D ). Then, the design parameters of the TID, D , will be selected by minimising a cost function ( D ). Function ( D ) can describe any statistics of the response, for example, the statistical moments or quantiles of engineering demand parameters (e.g., the maximum displacement or the interstorey drift) as functions of the response X(t; D ). For convenience, the notation of the dependence of the controlled system's response on D will not be carried on.

Structural system
We assume that the TID is installed in a nonlinear MDOF structure. Specifically, the example structure we consider is a N dof -DOF system with a cubic nonlinearity, whose response is represented by the vector 1), described by the following equation of motion: , N dof } are the mass, damping, and stiffness matrices; vector 1 is a unit vector of dimension (N dof , 1); parameter is a nonlinear parameter; and A(t) is a stochastic process that describes the seismic input to which the structure is subjected. The input A(t) will be described in detail in the next section. Note that the vector For the numerical example in this paper, the MDOF used has N dof = 5 DOF, and specific values for the structural system assumed are presented in the Appendix. Figure 1 shows the backbone curves of the first and the fifth DOF, that is, max t≥0 |X 1 (t)| and max t≥0 |X 5 (t)|, for which X(t) is the response of the structural system subjected to A(t) = a 0 sin( t). Note that the backbone curves are plotted against the ratio between the harmonic frequency and the dominant frequency of vibration of the structural system. As seen in Figure 1, the system exhibits a softening behaviour as the amplitude a 0 of the harmonic loading increases, which is typical for buildings. 63

FIGURE 1
Backbone curves for the first (DOF 1 ) and the fifth (DOF 5 ) DOF s

Tuned inetrer damper
The aim of this paper is to propose a novel, computationally efficient framework to perform a seismic performance-based design of passive vibration-control systems, and we demonstrate how to use this framework for the design of a TID installed in an MDOF nonlinear structure subjected to seismic loads, as defined in Equation 1. The TID, as already described in Section 1, is a passive vibration-suppression device, explored previously in civil engineering theoretical and experimental studies for the same purpose as other passive vibration-control systems, such as the VD, or the tunedmass dampers (TMD), to reduce vibrations in structures. The TID combines the advantages of both the VD and the TMD. 8 Like the TMD, the TID introduced an additional DOF to the structure, with additional damping and stiffness. Unlike the TMD, the TID's physical mass is replaced by an apparent mass produced by the inerter, and similar to the VD, the TID is connected to two terminals of the structure. Thus, the TID is able to reduce the system's response at the main frequency, similar to the TMD, and the response at the other secondary frequencies, similar to the VD. The TID as defined by Lazar et al., 8  The design of the TID consists of finding parameters D with respect to a performance criteria as a function of the structural system's response X(t). The apparent mass m D = D ∑N do i=1 m i,i is a characteristic of the inerter, which is commonly chosen by the designer when selecting the type of inerter used. A value of D = 20% is selected a priori to account for the capability of the inerter to generate an increased apparent mass. By comparison with the traditional TMD's mass that can go up to a maximum of 10% of the host structure's mass, the value D = 20% being only two times larger is reasonable compared with larger values reported in the literature. 9 Thus, the design of the TID reduces to the calculation of just two parameters ( D , D ), based on the performance criterion defined by the minimisation of a cost function ( D , D ). For the purpose of this paper, we define the cost function as where F X (x) = P{max t≥0 |X 5 (t)| ≤ x} is the probability distribution function of the maximum displacement of the controlled structure X(t), for a given set of parameters ( D , D ). Note that the maximum absolute displacement max t≥0 |X 5 (t)| can be replaced by any other engineering demand parameter, such as the maximum absolute interstorey The optimum design parameters ( D , D ) will be given by the minimum value of ( D , D ), over a range of values for ( D , D ). This range is chosen such that the damping and stiffness parameters remain realistic from a practical point of view. A maximum damping ratio D = 30% and a maximum frequency D = 8 rad∕s are assumed.

SEISMIC HAZARD
Large numbers of ground-motion records are required to perform probabilistic analyses for the response of structures to earthquakes. The seismic-hazard model used for the seismic performance-based design proposed in this study has two main components: (a) the distribution of earthquakes with respect to the moment magnitude m and the source-to-site distance r; and (b) the ground-motion model for the simulation of ground-motion time histories, as a function of (m, r), and other secondary parameters. The (m, r) distribution of the seismic hazard is essential in the seismic design of structures because the frequency content of the ground motions depends on (m, r), among other characteristics, such as local site conditions or seismic regime. 29,64 The distribution of earthquakes by (m, r) is obtained from the UHT 65 from the United States Geological Survey. This online tool provides the contribution of the seismic sources characterized by (m, r) for a specified level of the seismic intensity (e.g., the spectral acceleration) at each site in the United States. This (m, r)−distribution is also known as the seismic disaggregation defined in detail by McGuirre 66 and Bazzurro and Cornell. 67 For the numerical example in this study, downtown Los Angeles with a shear velocity in the top 30m of soil of v s30 = 760 m∕s is chosen as the site for the design of the TID. Note that in our methodology, we utilise the UHT's disaggregation only to extract the contribution of FIGURE 2 Seismic activity matrices for Los Angeles, for the probability of exceedance of (a) 10% and (b) 2% in 50 years, with v s30 = 760m∕s each source characterised by (m, r) at a given site. For convenience, we selected just (m, r)-earthquakes that correspond to a probability of occurrence equal to or larger than 10 −5 , which is achieved for m ≥ 5 and r ≤ 100 km. The normalised values of contributions of each pair (m, r) result in the probability mass function for the bivariate vector (M, R), where M and R are the random variables corresponding to the moment magnitude m and source-to-site distance r, also known as the seismic activity matrix. 23 We denote by P{(M, R) = (m, r)} the probability of occurrence of an earthquake characterised by (m, r). Figure 2 shows the seismic activity matrices for downtown Los Angeles, obtained from the UHT's disaggregation for a level of the hazard characterized by a probability of exceedance of (a) 10% and (b) 2% in 50 years, respectively.

Ground-motion simulation
The ground-motion model used in this study allows the simulation of synthetic earthquakes as a functions of (m, r), local soil conditions in term of v s30 and seismic regime. The model is based on the specific barrier model (SBM) 68,69 and the spectral representation methods. 54,70 The synthetic motions are samples of a nonstationary, non-Gaussian stochastic process A(t) with finite duration t f : where is a deterministic amplitude-modulation function with constant parameters , , and . The process Z(t) is a zero-mean, stationary, process with a Student's T marginal distribution. The second-order moment properties of Z(t) are given by the one-sided spectral density function g( ; m, r), ≥ 0 obtained from the SBM calibrated to global data 71 or calibrated to site records by Radu and Grigoriu. 23 Parameters , , in Equation 4 and the duration 0 ≤ t ≤ t f are also functions of (m, r), and are outputs of the SBM. The model in Equation 3 is a simplified version of the ground-motion model introduced by Radu and Grigoriu. 64 The advantage of using the model in Equation 3 is due to the fact that it preserves the realistic non-Gaussian character of earthquakes, a feature usually not considered in other models. 31,72 The non-Gaussianity of real earthquakes is supported by the results in Figure 3a, which show that the kurtosis coefficient as a function of v s30 for the ground motions in the NGA-West data set 73 is greater than 3, the characteristic value for Gaussian processes. Figure 3b shows the probability distribution function of the kurtosis for the type-C National Earthquake Hazard Reduction Program (NEHRP) soil, similar to the soil assumed for this study. The distribution of Z(t) is calibrated to the average kurtosis coefficient corresponding to type-C NEHRP soil C = 14.3 > > 3 as shown by Radu and Grigoriu. 64 Figure 4a illustrates the power-spectral density functions g( ; m, r) for three pairs of (m, r) = (6.3, 30 km), (m, r) = (7.5, 70 km), and (m, r) = (8.1, 110 km), for v s30 = 760 m∕s, which shows that the frequency content of the motion is dependent on (m, r). Figure 4b shows time-history samples of A(t) in Equation 3 for the three pairs of (m, r) in (a), with their corresponding frequency contents and lengths t f .  1, 110 km), respectively Other ground-motion stochastic simulation models such as the ones proposed by Rezaeian and Kiureghian 31,72 or Vlachos et al., 74 or more complex physics-based models such as Goda et al., 75 can be used to simulate earthquakes as functions of (m, r), and so can be substitutes of the model adopted in this study, within our proposed framework. Irrespective of whether the substitute model is fully defined or just its samples are available, it suffice to just build the respective SROM based on sample statistics.

SROM for seismic input
As discussed before, a large number of samples of X(t), for each pair of the TID parameters ( D ; D ), are necessary to obtain reliable values for the design metric in Equation 2. This would imply a significant computational effort, because each sample of X(t) is obtained through the solution of Equation 1 for a sample of A(t). Thus, in order to resolve this issue, an SROM for the earthquake process A(t) is constructed. An SROMÃ(t) of A(t) is defined by any numberñ of samplesã k (t) of A(t) with probabilities p k such that ∑ñ k=1 p k = 1. Similar to MC, the SROM also uses random samples of A(t), but unlike in the case of MC, the samples are not equally likely but weighed by distinct probabilities. Pairs of A(t) samples and their probabilities, {(ã k (t), p k ), k = 1, . … ,ñ}, are selected such that A(t) and A(t) have similar probability laws. It was shown by Grigoriu 60,76 that it is possible to select a relatively small number n of independent samples of A(t) to construct an SROMÃ(t). Our objective is to find p k such that the discrepancies between the probability laws of the SROMÃ(t) = {(ã k (t), p k ), k = 1, … ,ñ} and A(t) are minimized. The probability laws of stochastic processes are defined by their moments, marginal distributions, and covariance functions. 77 The target marginal distribution F(x; t), moments (t; q) of order q, and covariance function Σ(t, s) of A(t) can be calculated directly from its samples or directly from the definition in Equation 3. The same statistics for the SROM A(t) are calculated from itsñ samplesã k (t), weighed by their probabilities p k . Thus, the moments̃(t; q) of order q, the marginal distributionF(x; t), and the covariance functionΣ(t, s) of the SROMÃ(t) are calculated respectively as follows:   Figure 5 shows (a) the MC estimators of the marginal distribution of A(t) calculated with n = 5, 000 samples and the respective approximate SROM estimatorsF(x, t) ofÃ(t) for (b)ñ = 10 and (c)ñ = 20, for A(t) calculated for (m, r) = (6.3, 30km). Panels (a) and (b) of Figure 6 show the first two-order moments, that is, (t, 1) and (t, 2), of the process A(t) calculated by the MC estimators with n = 5, 000 samples and for the SROMÃ(t), withñ = 10 andñ = 20, respectively. Finally, Figure 7 illustrates (a) the MC estimator of the covariance function Σ(t, s) of A(t) calculated with n = 5, 000 samples and the SROM estimatorsΣ(t, s) ofÃ(t) for (b)ñ = 10 and (c)ñ = 20, respectively.
In Figures 5-7, it can be noticed that all SROM statistics are improving as the dimensionñ increases from 10 to 20, a value for which they approach the MC target statistics within a reasonable accuracy. A further discussion on the topic of the SROM accuracy is presented in the following section. More importantly, a sensitivity analysis on how the MC estimates of ( D ; D ) in Equation 2 compare with the SROM approximates is also provided. As shown herein, the SROM is built based on the target samples statistics of the input, independent of the system subjected to it, and thus, the system used as an example in Equation 1 can be replaced by any other customary structure.

Sensitivity analysis
The performance ofÃ(t) depends on the SROM's sizeñ, and for a givenñ, the accuracy of the SROM depends on its construction, that is, the selection of samplesã k (t) and their probabilities p k . Thus, increasing the dimension of the SROM n → ∞ would produce very accurate statistics of the process A(t), as it converges to the MC solution, 59,78 but it comes at a large computational cost. However, the real gain of using SROMs can only be judged with respect to the response statistics, mainly the metric ( D ; D ) in Equation 2 used for the design of the TID's parameters ( D ; D ). The accuracy of the response statistics for X(t) depends on that ofÃ(t) and the assumption that samples ofã k (t) map into samples ofx k (t) with same probabilities p k , by solving Equation 1. Figure where FX (x) = ∑ñ k=1 p k 1{max t≥0 |x k,1 (t)| ≤ x} is the SROM estimator for the response probability distribution function. Figure 9 shows the contour plot of the MC estimate of ( D , D ) from Figure   previous benchmark studies of the TID, two additional values of ( D , D ) corresponding to the deterministic harmonic approach developed in the study of Lazar et al., 8 and to Den Hartog's approach, 5 are illustrated.
It is important to notice in Figure 9 that the SROM estimates of ( D , D ) are reliable and that a SROM dimension of n = 20 produces results nearly identical with the MC estimate. The errors of SROM statistics are bounded with bounds established by Warner et al. 76 and Uy and Grigoriu, 62 but this analysis is beyond the scope of this paper.

FULL SITE-SEISMICITY DESIGN
The previous section established the effect of TID parameter selection on the structural response to a (m, r) = (6.3, 30km) earthquake. We now extend our study to consider the full range of (m, r) values in the SAM that characterises the chosen site as established in Section 3. The design methodology for TIDs proposed in this study takes into the account that the seismic ground-motions' frequency contents are functions of (m, r) as shown in Figure 4 and that the structural response is highly sensitive to this frequency content. Figure 10 shows the design parameters ( D , D ) obtained using the stochastic estimator in Equation 8, compared with the harmonic and Den Hartog deterministic estimates for three different types of earthquakes: (a) (m, r) = (6.3, 30km); (b) (m, r) = (6.7, 30km); (c) (m, r) = (8.1, 110km). The deterministic approaches are insensitive to the types of earthquakes, whereas the TID design parameters can vary significantly with respect to (m, r). In most of the cases tested, the frequency estimates D do not vary significantly from the deterministic values. However, large discrepancies may be noticed in the damping ratios D with respect to the deterministic results, which in the most of the cases are conservative. Given that only one value of ( D , D ) can be selected for the design of a TID, the (m, r)-dependence of ( D , D ) is impractical. Thus, for a comprehensive design of the TID with respect to the seismicity at a site, a SROMÃ(t) must be constructed, and̃m ,r ( D , D ) in Equation 8 must be calculated for each (m, r) in the SAM (see Figure 2). The values for the design of the TID will be calculated by minimizing the cost function SAM ( D , D ) defined for the entire site seismicity, that is, for the entire SAM:  Figure 2. The bold dashed line shows the mean function 1−FX (x) calculated by using the mean TID parameters for each (m, r) in the SAM, for which P{(M, R) = (m, r)} > 0, that is, assuming that all earthquakes in the SAM are equally likely to occur. This shows that simply taking the average of ( D , D ) over (m, r) can result in significant underestimating the response of the structure.
To assess further these results, the following analysis of the TID-controlled system subjected to the real ground-motion records in the NGA-West data set 73 is now considered. This data set contains approximately 3,500 time histories of different magnitudes, source-to-site distances, and recorded at various sites characterized by different values of v s30 . Figure 12 shows scatter plots for each record in the data set that has a maximum absolute displacement of the uncontrolled system greater than or equal to 0.005, for the case where the optimal ( D , D ), we selected using the proposed stochastic approach and the harmonic and Den Hartog approaches, as shown in Figure 11.  Figure 11a, that is, the SROM stochastic approach, and the deterministic harmonic and Den Hartog approaches are tested for the design of the TID. Note that the points above the dashed lines show records for which the TID-controlled system underperforms with respect to the uncontrolled system, that is, fails to provide a lower maximum absolute displacement. Figure 12b shows the percentage differences between the maximum absolute displacements of the uncontrolled and TID-controlled systems, normalised by the uncontrolled displacements, for all three TID designs as in panel (a). Note also that a negative value, that is, a point below the dashed line, illustrates a record for which the TID-controlled system underperforms when compared with the uncontrolled system. The solid lines in Figure 12b are regression lines for the percentage differences for each of the three TID designs analysed.
Qualitatively, from Figure 12, it can be inferred that the stochastic and the harmonic designs of the TID perform similarly, for the system and site considered in this study. Furthermore, the reduction in the structural response through these two designs of the TID is larger than in the case of the Den Hartog design, as it is seen in both plots. Quantitative results regarding the stochastic and the two deterministic designs of the TID for the NGA-West analyses shown in Figure 12 are presented in Table 1. This table, as well as Figure 11a, shows that all three designs of the TID have notable differences in the damping ratios D , for similar frequency values D . More specifically, the differences in the performances of the TID-controlled systems are further expressed as the average percentage of successes, that is, the ratio between the number of records for which the TID-controlled systems exhibit lower maximum absolute displacements than the uncontrolled systems. Another metric used for this comparison is the average damage reduction, which is the mean ratio between the maximum absolute displacements of the TID-controlled system and the uncontrolled system.
Whereas the Den Hartog design of the TID-controlled system fails to provide improved responses only in 3.5% of the cases, the other two designs fail in approximately 9% of the cases but provide better responses by reducing the maximum absolute responses by approximately 15%, rather than only 10% in the case of the Den Hartog design. Finally, the stochastic framework proposed for the design of the TIDs is able to provide rigorous values consistent with the seismic hazard at a site. However, in the current example, of the seismic hazard in Los Angeles and the MDOF structure considered, the stochastic and the deterministic harmonic designs provide TID designs with similar performances. Nevertheless, the performance-based framework proposed does provide a better understanding on how the TID performs with respect to different earthquakes, inputs with complex various frequency contents, and it can make a significant difference as, for  example, in the case shown in Figure 9b, where the damping ratio D of the TID is only 2%, rather than 13% as in the case of the harmonic design. Moreover, this study provides a highly efficient framework for the stochastic design of the TID that provides reliable robust results with a computational effort smaller by a factor of 250 than the same result obtained by MC, or in absolute values, for a SAM of 200 (m, r) pairs as in the example herein, the number of dynamic analyses is reduced from one million to only 4,000.

CONCLUSIONS
A novel methodology based on SROMs is proposed for the fully probabilistic design of TIDs with respect to their performance in civil engineering structures subjected to seismic loads. The framework is general, and the TID is designed for nonlinear MDOF structures, subjected to synthetic nonstationary, non-Gaussian seismic ground motions. The design parameters of the TID using the methodology proposed are consistent with the classical methodology for probabilistic design, based on MC simulations. The design framework proposed is computationally efficient, robust, and easy to use. This method allows a fully probabilistic design of the TID to account for the site seismicity, by accounting for the character of ground motions with respect to seismic-source and local site characteristics. Finally, results obtained are compared with deterministic benchmark approaches, which shows that the probabilistic design may lead to more economical results for certain types of earthquakes, while preserving the performance of the more conservative deterministic approaches currently used in control engineering.
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APPENDIX A
This appendix provides information about the uncontrolled and TID-controlled structures in Sections 2.1 and 2.2, respectively. The structure with a TID mounted at the ground floor, as assumed in this study, is illustrated schematically in Figure A1. The structure considered in this study has N dof = 5 degrees of freedom. The original uncontrolled structure is described by the mass M = {m i,j |m i,i = 1; m i,j = 0, i ≠ j; i, j = 1, … , 5}, damping C = {c i,j |c i,i = 6.34, i ≤ 4; For the TID-controlled system, the mass M, damping C, and stiffness K matrices in the equation of motion Equation 1 are replaced by M (d) , C (d) , and K (d) of the structural system equipped with the TID. Note that the new matrices for the TID-controlled structure have N dof + 1 = 6 degrees of freedom, the additional one being introduced by the TID, and are described in detail in Section 2.2. For a better understanding of how the TID affects the behaviour of the host structure, the matrix forms of M (d) , C (d) and K (d) are provided below as functions of the original mass, damping and stiffness matrices M, C and K: The updated stiffness matrix has the same layout as C (d) , where c d is replaced by k d and C is replaced by K.