Seismic Performance Assessment of Optimal Tandem-Based Tuned Mass Damper Inerters

Abstract

In the current work, two novel tandem-based tuned mass damper configurations are introduced. These configurations extend the recently proposed tuned tandem mass damper inerter (TTMDI) by replacing the linking dashpot with an inerter (i.e., the inerter-connected TTMDI (ICTTMDI)), and an integrated tuned tandem mass damper inerter (I-TTMDI) by integrating recently proposed tuned tandem mass damper (TTMD) configurations. The control efficiency of the optimally designed dampers for a single-degree-of-freedom (SDOF) system was evaluated in a uniform framework to reveal and compare the performances of the ICTTMDI and I-TTMDI with those of other recently proposed tandem-based configurations. The optimum design of all the studied configurations was determined by the particle swarm optimization (PSO) algorithm. The evaluation of the performances included the effectiveness in the frequency domain and that of the norm and maximum reduction in the displacement and absolute acceleration in the time domain under 21 earthquake records with different characteristics. Additionally, the strokes of the dampers, the structure energies, and the power spectral densities (PSDs) of the responses were investigated. The optimum design of the I-TTMDI revealed the best configuration by determining the optimum distributions of the mass and inertance between the tandem mass and inerter links, respectively. The proposed configuration not only demonstrated improved response reduction across the displacement and acceleration measures but also maintained remarkable robustness under 21 earthquake records (far-fault, near-fault forward-directivity, and fling-step records). Furthermore, the advantages of the side inerter distribution were particularly effective at widening the operating frequency band, breaking through the traditional limitations of TMD-based devices. The consistent performances of the newly proposed configurations prove that they can be used to advance the development of more reliable structural control systems.

1. Introduction

Tuned mass dampers (TMDs) and their derivatives are among the most widely studied and employed passive control devices for the absorbance of the dynamic vibrations of different buildings [1,2,3,4,5,6], bridges [7,8,9], wind turbines [10,11,12,13], offshore platforms [14], high-rise chimneys [15], transmission lines [16], historical monuments [17], and other structures [18,19,20,21,22,23,24,25,26,27,28,29]. The promising features of TMDs render them a hot topic for research [18,30] and reliable devices for practical application [31] in the structural control community. However, to mature TMDs for seismic vibration control, their main shortcomings, i.e., their narrowband effectiveness, high detuning sensitivity, and the dependency of their performance on the excitation frequency content, should be resolved [32]. Therefore, in the past decade, innovative modifications have been proposed to eliminate the aforementioned deficiencies [30,33,34,35,36]. One of the most preeminent achievements is the tuned mass damper inerter (TMDI) by Marian and Giaralis [37]. Consequently, several studies have been carried out to improve the efficiency of TMDs [38,39,40,41]. Recently, the tuned tandem mass damper (TTMD) was introduced by Yang and Li [42] to improve the seismic performance of the TMD. Accordingly, Cao and Li [43] mixed these two control devices and established the tuned tandem mass damper inerter (TTMDI), which is aimed at improving the tandem arrangement by employing an inerter device. The most recent tandem configuration is proposed as the inerter-connected tuned tandem mass damper (ICTTMD) and was assessed by Djerouni et al. [44] and Hojat Jalali and Fahimi Farzam [45]. Figure 1 summarizes and illustrates the schematic representations of the proposed configurations. Previous research has shown that the introduction of inertance in the TMDI is accompanied by two opposite characteristics: negative stiffness and mass amplification [46,47]. In the case of the grounded TMDI (GTMDI), the presence of the inerter improves the response due to pure mass amplification effects. However, in other cases wherein the inerter is not directly grounded, the inerter can have negative/positive effects on the performance of the control device depending on which one prevails. Negative stiffness reduces the efficiency of the ICTTMD, while mass amplification enhances its performance. Furthermore, increasing the inertance (b) can exacerbate both the negative stiffness and mass amplification effects [47,48].

To fully exploit the classical TMD, usually, an optimization problem is defined with a set of dimensionless optimization variables, i.e., mass, frequency, and damping ratios [49]. Previous studies have ascertained that the increase in the mass ratio improves the control performance of the TMD [50]; however, it is somewhat trivial that the mass ratio has a practical upper bound. Therefore, the mass ratio is conventionally preselected based on construction limitations, while the damping and frequency ratios are assumed to be optimization variables [51].

Due to the similarity between inertance and mass, a similar approach has been used in studies of inerter-based control devices, and the amount of inertance is often preselected [43]. Although this is an accepted method in the conventional TMD and TMDI configurations, in the tandem-based configurations, the distributions of the total mass and inertance between the masses and inerters can have a profound effect on the performance of the damper [42,43,44]. Hence, the optimal mass and inerter distributions not only improve the performance of the proposed configuration but also determine the optimal configuration itself. However, in the previous studies, the effects of the mass and inertance distributions were studied based on preselected distribution ratios and not as an optimization problem [42,43]. Hence, the determination of the best configuration of all possible tandem-based configurations requires comparing all configurations under the same design and evaluation framework.

In order to address the abovementioned concerns, two new tandem-based configurations are proposed as modifications to the TTMDI (Figure 2). The first newly proposed configuration extends the TTMDI by replacing the dashpot with an inerter (i.e., the ICTTMDI (Figure 2a)), and the second integrated configuration, the I-TTMDI, not only represents all the previously presented configurations but also includes other possible tandem-based configurations (Figure 2b). In the newly introduced configurations, the mass and inertance ratios are also assumed to be part of the optimization variables, in which the total mass and inertance are optimally distributed in the process. Therefore, the most efficient configurations without any predefined constraints on the mass and inertance distributions can be achieved in a uniform optimization and evaluation framework.

To this end, the free design parameters of the tandem-based configurations for the passive control of a single-degree-of-freedom (SDOF) structure were obtained using the PSO algorithm for two groups of configurations (i.e., with the predefined distribution and free distribution of the mass and inertance ratios) based on the H₂ norm of the transfer function of the main mass displacement. Furthermore, the performances of the newly proposed configurations (i.e., the ICTTMDI and I-TTMDI) under 21 benchmark ground motions were assessed through different performance criteria, such as the norm and maximum of the displacement and the absolute acceleration of the main mass, the strokes of the TMDs, the structure energies, and the power spectral densities (PSDs) of the structural responses. Finally, the results were compared to those of the classical TMD, TMDI, and recently proposed tandem-based configurations. In the remainder of this study, the TMD, TTMD, and ICTTMD are referred to as TMD-based devices, while the TMDI, ICTTMDI, and I-TTMDI are identified as TMDI-based devices.

2. Materials and Methods

In order to present the fundamental performances of the proposed configurations and to provide a basis for comparison with previous studies [42,43,52], an SDOF structure with ground access was selected as the host structure in this study. Although, for the control devices on low-rise structures or base-isolated systems, the second terminal of the inerter has access to the ground, the proposed configurations are not readily applicable to high-rise structures. In these cases, researchers have investigated providing span-story installations [48,53,54,55,56], where the second terminal of the inerter spans more than one story. Nonetheless, investigating the possibility of ground-connected inerters in multi-story structures is outside of the scope of this study.

In the current optimization framework, eight different configurations, including six tandem-based configurations and two classical TMD and TMDI devices, were optimally tuned with the same total mass and inertance ratios. In

Table 1. Optimization variable definitions and variable ranges.

Models	Design Variables
	Preselected Variables	Optimization Variables
Configurations with pre-distributed mass and inertance
TMD	TMR	fd1, ζd1
TMDI	TMR, TIR	fd1, ζd1
ICTTMD	TMR, RMR, TIR	fd1, fd2, ζd1, ζd2
TTMD	TMR, RMR	fd1, fd2, ζd1, ζd2, ζdc
TTMDI	TMR, RMR, RIR, TIR	fd1, fd2, ζd1, ζd2, ζdc
Configurations with free-distributed mass and inertance
TTMDI-F	TMR, TIR	fd1, fd2, ζd1, ζd2, ζdc, RMR, RIR
ICTTMDI-F	TMR, TIR	fd1, fd2, ζd1, ζd2, RMR, RIR, RIRc
I-TTMDI-F	TMR, TIR	fd1, fd2, ζd1, ζd2, ζdc, RMR, RIR, RIRc
Variable definitions and their ranges	$\mathrm{T}\mathrm{M}\mathrm{R} (\mu )={\displaystyle \frac{m_d}{\mathrm{M}}}=0.01$ $\mathrm{R}\mathrm{M}\mathrm{R}={\displaystyle \frac{{\mathrm{\mu }}_1}{{\mathrm{\mu }}_2}}={\displaystyle \frac{m_{d1}}{m_{d2}}}=1,0.75$ $TIR(\beta )={\displaystyle \frac{b}{\mathrm{M}}}$ = 0.05 SDOF mass: M = 1.11 × 106 kg SDOF frequency: ${\omega }_s=\sqrt{{\displaystyle \frac{K}{M}}}=3.14$ rad/s SDOF inherent damping: ${\mathsf{\zeta }}_s={\displaystyle \frac{C}{2M{\omega }_s}}=0.02$	$f_{d1}={\displaystyle \frac{{\omega }_{d1}}{{\omega }_{1s}}}={\displaystyle \frac{\sqrt{\frac{k_{d1}}{m_{d1}+b_1+b_c}}}{{\omega }_{1s}}}$ ${\mathsf{\zeta }}_{d1}={\displaystyle \frac{c_{d1}}{2(m_{d1}+b_1+b_c){\omega }_{d1}}}$ $f_{d2}={\displaystyle \frac{{\omega }_{d2}}{{\omega }_{1s}}}={\displaystyle \frac{\sqrt{\frac{k_{d2}}{m_{d2}+b_2+b_c}}}{{\omega }_{1s}}}$ ${\mathsf{\zeta }}_{d2}={\displaystyle \frac{c_{d2}}{2(m_{d2}+b_2+b_c){\omega }_{d2}}}$ ${\mathsf{\zeta }}_{d2}={\displaystyle \frac{c_{d2}}{2(m_{d1}+m_{d2}+b_c){\omega }_{av}}}$ ${\omega }_{av}={\displaystyle \frac{{\omega }_{d1}+{\omega }_{d2}}{2}}$ $RMR={\displaystyle \frac{{\mathrm{\mu }}_1}{{\mathrm{\mu }}_2}}={\displaystyle \frac{m_{d1}}{m_{d2}}}$ $RIR={\displaystyle \frac{{\beta }_1}{{\beta }_2}}={\displaystyle \frac{b_{d1}}{b_{d2}}}$   ${\mathrm{R}\mathrm{I}\mathrm{R}}_{\mathrm{c}}={\displaystyle \frac{{\beta }_c}{{{\beta }_1+\beta }_2}}={\displaystyle \frac{b_{d1}}{b_{d2}}}$ fd1, fd2= [0.01 1.5], ζd1, ζd2, ζdc = [0 1] RMR, RIR, RIR c = [0 1]

, these configurations, their preselected and design variables, the definitions of these variables, and their search domains are presented, where m_d1, c_d1, k_d1, and b_d1 (m_d2, c_d2, k_d2, b_d2) denote the mass, damping, stiffness, and inertance of the first (second) TMD, respectively; moreover, b_c and Cc are the connecting inertance and damping, respectively; similarly, M, C, and K stand for the mass, damping, and stiffness of the uncontrolled SDOF structure, respectively. Therefore, the differential equation of motion for a tandem-based controlled structure subjected to ground acceleration can be written as follows:

$$\mathbf{M}\ddot{\mathbf{Y}}\left(\mathbf{t}\right)+\mathbf{C}\dot{\mathbf{X}}\left(\mathbf{t}\right)+\mathbf{K}\mathbf{X}\left(\mathbf{t}\right)=\mathbf{0}$$

(1)

where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix of the controlled structure, and Y(t) is the absolute displacement of the controlled structure, which is the sum of the relative displacement (X(t)) and ground displacement (X_g(t)). A dot above the variable represents the derivative of the variable with respect to time.

Three distribution design variables, i.e., the relative mass ratio (RMR), relative inertance ratio (RIR), and relative connecting inertance ratio (RIR_c), regulate the distributions of the total mass ratio (TMR) and total inerter ratio (TIR) between two tandem masses, two side inerters, and the connecting and side inerters, respectively. In Table 1, to indicate the possibility of the free and unconditional distributions of the mass and inertance in the last three devices, “-F” has been added to the end of the abbreviated names of these three control devices. Based on Table 1, the I-TTMDI-F has ten design variables, compared to three for the classical TMD and seven for the TTMD, for which a more flexible design would possibly lead to better results.

In the I-TTMDI-F, the total mass and inertance ratios (i.e., the TMRs and TIRs in Table 1) were the only preselected design parameters based on the practical restrictions, and the remaining eight parameters, (i.e., f_d1, f_d2, ζ_d1, ζ_d2, ζ_dc, RMR, RIR, and RIR_c in Table 2) were assumed to be the design variables for the optimization problem of the newly introduced dampers. The last row of Table 1 shows the host SDOF system and the aforementioned variables. The values for the SDOF system correspond to the first mode of vibration of a 10-story benchmark building reported in the literature [32,45,57,58].

Notably, for comparing the performances of the proposed dampers with the suggested distributions of mass, three tandem-based devices with pre-divided masses were also investigated for comparison purposes. In each case, two configurations with the preselected RMR values of 1 and 0.75 and an equal distribution of inertance between the inerters were modeled [42,43]; however, in the newly proposed ICTTMDI-F and I-TTMDI-F and the extended form of the TTMDI (i.e., TTMDI-F), the RMR, RIR, and RIRc are optimization variables, and their optimum values are determined in the optimization process.

Due to the stochastic nature and broadband frequency content of earthquake records, especially under near-fault excitations, the H₂-norm minimization was adopted in this study for determining the optimal parameters of the dampers to provide a robust design and ensure the proper performances of the proposed control devices. The objective of the H₂ optimization was to reduce the total vibration energy of the system over all frequencies by minimizing the area under the frequency response curve. This is also recognized by different researchers, who preferred the H₂ to H_∞ minimization for optimizing dampers under stochastic and broadband frequency excitations [44,52,59,60,61,62]. However, other robust design methods, such as minimizing the H∞ norm, can be considered and compared in a separate study but are not considered in the current work.

The selected objective function for the optimization problem was minimizing the H₂ norm of the maximum displacement of the main mass as an overall performance index of the control devices. This objective function (Equation (2)) is a 2-norm of a single-input single-output (SISO) transfer function of the combined host building and TMD-based control configuration:

$${‖\mathbf{H}‖}_{\mathbf{2}}=\sqrt{\left[{\displaystyle \frac{\mathbf{1}}{\mathbf{2}\mathsf{\pi }}}{\int }_{-\mathit{\infty }}^{+\mathit{\infty }}{\mathbf{H}}^{\mathit{*}}\mathbf{H}\mathit{d}\mathit{\omega }\right]}$$

(2)

where H is the transfer function that relates the input and output of the system. Once the optimal parameters of the control devices were obtained and assessed in the frequency domain, time history analyses were carried out to evaluate the performances of the tandem-based configurations in the time domain, as further elaborated in Section 4. To this end, three sets of seven benchmark natural ground motions of far-fault (FF) and near-fault (NF) earthquakes with forward-directivity (FD) and fling-step (FS) characteristics were employed to include the uncertainty in the frequency contents of the earthquake records. The records and relevant information are shown in Table 2. The evaluation of the damper efficiency in the time domain was based on the maximum and norm of the main mass displacement and absolute acceleration, the PSDs of these responses, the strokes of the TMDs, and the structure energies under earthquake records.

Table 2. Seismic record information used in this study [63].

No.	Year	Earthquake	Station	Com.	Source	PGA (g)	PGV (cm/Sec)	PGD (cm)	Fling Disp. (cm)
(a) Far-Fault Records
1	1952	Kern County	Taft	111	1	0.18	17.5	8.79	N.A.
2	1979	Imperial Valley	Calexico	225	1	0.27	21.24	9.03	N.A.
3	1989	Loma Prieta	Presidio	0	1	0.1	12.91	4.32	N.A.
4	1994	Northridge	Century CCC	90	2	0.26	21.19	7.85	N.A.
5	1994	Northridge	Moorpark	180	2	0.29	20.97	5.48	N.A.
6	1994	Northridge	Montebello	206	1	0.18	9.41	1.51	N.A.
7	1971	San Fernando	Castaic	291	1	0.27	25.9	4.87	N.A.
(b) Near-Fault Records (Forward-Rupture Directivity)
8	1992	Cape Mendocino	Petrolia	90	1	0.66	90.16	28.89	N.A.
9	1994	Northridge	Olive View	360	1	0.84	130.37	31.72	N.A.
10	1992	Erzincan	Erzincan	EW	1	0.5	64.32	21.93	N.A.
11	2004	Parkfield	Fault Zone 1	FN	5	0.5	64.15	12.64	N.A.
12	1984	Morgan Hill	Anderson Dam	340	2	0.29	28	12.19	N.A.
13	1987	Superstition Hills	Parachute Test Site	315	1	0.45	112	52.46	N.A.
14	1979	Imperial-Valley	Brawley Airport	225	1	0.16	35.85	22.39	N.A.
(c) Near-Fault Records (Fling Step)
15	1999	Kocaeli	Yarimca (YPT)	EW	3	0.23	88.83	184.84	145.79
16	1999	Chi-Chi	TCU052	NS	4	0.44	216	709.09	697.12
17	1999	Chi-Chi	TCU068	EW	4	0.5	277.56	715.82	601.84
18	1999	Chi-Chi	TCU074	EW	4	0.59	68.9	193.22	174.56
19	1999	Chi-Chi	TCU084	EW	4	0.98	140.43	204.59	161.82
20	1999	Chi-Chi	TCU102	EW	4	0.29	84.52	153.88	73.66
21	1999	Chi-Chi	TCU128	EW	4	0.14	59.42	91.05	49.88

N.A.: Not Applicable.

Table 2. Seismic record information used in this study [63].

No.	Year	Earthquake	Station	Com.	Source	PGA (g)	PGV (cm/Sec)	PGD (cm)	Fling Disp. (cm)
(a) Far-Fault Records
1	1952	Kern County	Taft	111	1	0.18	17.5	8.79	N.A.
2	1979	Imperial Valley	Calexico	225	1	0.27	21.24	9.03	N.A.
3	1989	Loma Prieta	Presidio	0	1	0.1	12.91	4.32	N.A.
4	1994	Northridge	Century CCC	90	2	0.26	21.19	7.85	N.A.
5	1994	Northridge	Moorpark	180	2	0.29	20.97	5.48	N.A.
6	1994	Northridge	Montebello	206	1	0.18	9.41	1.51	N.A.
7	1971	San Fernando	Castaic	291	1	0.27	25.9	4.87	N.A.
(b) Near-Fault Records (Forward-Rupture Directivity)
8	1992	Cape Mendocino	Petrolia	90	1	0.66	90.16	28.89	N.A.
9	1994	Northridge	Olive View	360	1	0.84	130.37	31.72	N.A.
10	1992	Erzincan	Erzincan	EW	1	0.5	64.32	21.93	N.A.
11	2004	Parkfield	Fault Zone 1	FN	5	0.5	64.15	12.64	N.A.
12	1984	Morgan Hill	Anderson Dam	340	2	0.29	28	12.19	N.A.
13	1987	Superstition Hills	Parachute Test Site	315	1	0.45	112	52.46	N.A.
14	1979	Imperial-Valley	Brawley Airport	225	1	0.16	35.85	22.39	N.A.
(c) Near-Fault Records (Fling Step)
15	1999	Kocaeli	Yarimca (YPT)	EW	3	0.23	88.83	184.84	145.79
16	1999	Chi-Chi	TCU052	NS	4	0.44	216	709.09	697.12
17	1999	Chi-Chi	TCU068	EW	4	0.5	277.56	715.82	601.84
18	1999	Chi-Chi	TCU074	EW	4	0.59	68.9	193.22	174.56
19	1999	Chi-Chi	TCU084	EW	4	0.98	140.43	204.59	161.82
20	1999	Chi-Chi	TCU102	EW	4	0.29	84.52	153.88	73.66
21	1999	Chi-Chi	TCU128	EW	4	0.14	59.42	91.05	49.88

N.A.: Not Applicable.

3. Optimum Design of the Control Devices

The optimum design of TMDs has been at the center of different researchers’ interests since its introduction in the literature. Although many different analytical, quasi-analytical, and even heuristic methods have been implemented for the optimum design of TMDs, the metaheuristic algorithms are the only well-appreciated methods to find the optimum values of the design parameters for structures built with inherent damping [64,65]. Nevertheless, for the I-TTMDI-F, the number of design variables (frequency and damping ratios for the two TMDs) has increased notably. Thus, an appropriate metaheuristic algorithm should be selected for this complex optimization problem. PSO is a well-explored optimization algorithm in the optimum design of the TMD and its different forms [42,66,67]. Therefore, in this study, PSO was selected as the suitable algorithm for the optimization process. Further details on the PSO algorithm framework can be found in the literature [68,69,70].

In this study, the optimum free vibration parameters for the preselected mass ratio (μ = 1%) and inertance ratio (β = 5%) were calculated for the TMD, TMDI, and six tandem-based configurations for an SDOF structure with an inherent damping of 2%.

Table 3. Optimal parameters for different control devices with pre-distributions of 1% for the mass ratio and 5% for the inertance ratio.

Models	Pre-Distributed Parameters					Optimum Parameters					JF1
	βc	β1	β2	µ1	µ2	fd1	ζd1	fd2	ζd2	ζdc
TMD	-	-	-	0.01	-	0.9819	0.0498	-	-	-	0.7045
TMDI	-	0.05	-	0.01	-	0.9486	0.1198	-	-	-	0.5133
TTMD-1:1	-	-	-	0.005	0.005	0.9118	0	1.0583	0.0015	0.0222	0.6782
TTMD-3:4				0.0043	0.0057	0.9148	0	1.0434	0.0141	0.0172	0.6814
TTMDI-1:1	-	0.025	0.025	0.005	0.005	1.1326	0.0269	0.8292	0	0.2523	0.4898
TTMDI-3:4				0.0043	0.0057	1.1393	0.0225	0.8282	0	0.2619	0.4889
ICTTMD-1:1	0.05	-	-	0.005	0.005	0.0117	0.4207	0.4109	0.0104	-	0.7071
ICTTMD-3:4				0.0043	0.0057	0.1133	0.0885	0.3976	0	-	0.7048

and

Table 4. Optimal parameters for different control devices with free distributions of 1% for the mass ratio and 5% for the inertance ratio.

Models	Optimum Parameters								JF1
	RIRc (βc, β1 + β2)	RIR (β1, β2)	RMR (µ1, µ2)	fd1	ζd1	fd2	ζd2	ζdc
TTMDI-F	-	(2.3, 2.7)	(0.2, 0.8)	1.18	0	0.8281	0	0.3087	0.4851
ICTTMDI-F	(0.25, 4.75)	(2.38, 2.37)	(0.05, 0.95)	1.013	0.0962	0.8705	0.0514	0	0.5033
I-TTMDI-F	(0, 5)	(2.48, 2.52)	(0.001, 0.999)	1.1846	0	0.8273	0	0.3111	0.4834

present the optimal values of the design variables for different pre-distributed and free-distributed configurations, respectively, based on minimizing the H₂ norm of the main mass displacement using the PSO algorithm. The normalized H₂ norm of the displacement transfer function of the main mass, defined by Equation (3), is also reported for comparison purposes in the last columns of Table 3 and Table 4. Table 3 clearly shows that the pre-divided mass and inertance ratios for the TTMD, TTMDI, and ICDTMD are identical to the suggested values for the RMR in the published literature [42,43], where “−a:b” at the end of each model name indicates the RMR:

$$JF1={\displaystyle \frac{{‖H‖}_{2,Controlled}}{{‖H‖}_{2,Uncontrolled}}}$$

(3)

On the contrary, in Table 4, the distributions of the mass and inertance ratios for the newly proposed configurations were determined by the optimization algorithm. According to the calculated normalized two-norm of the displacement transfer function of the main mass, it can be observed that the I-TTMDI-F outperformed all the previously studied configurations.

A schematic presentation of the optimally designed tandem configurations with the free distributions of the masses and inertances is presented in Figure 3, where the dimensions of the masses and inertances are shown qualitatively in proportion to the optimally distributed values.

The free distributions of the masses and inertances show the redundancy of the connecting inerter between the two masses. This can possibly be attributed to the fact that by increasing the inertance of the connecting inerter between the two masses, the negative stiffness dominates the mass amplification effects, and, therefore, increasing the inertance between the two tandem masses reduces the performance of the control device. Particularly, the results of the optimization of the free-distribution models show that in the I-TTMDI-F configurations, the connecting inerter is practically removed (Figure 3a), and it is significantly reduced in the ICTTMDI-F configuration (Figure 3b). However, in all configurations, the side inerters share an equally distributed inertance ratio, and the distribution of the TMR shows a tendency for a completely unbalanced distribution. Notably, according to the values of the optimized masses and inertances, an equivalent mass $(m_{ei}=m_{di}+b_i, i=1, 2)$ corresponding to each TMD_i (i = 1, 2) can be defined, with the optimal ratio ($m_{e1}/m_{e2})$ almost constant and equal to 0.7 in all the tandem-based devices either with or without side and connecting inerters. Remarkably, the optimum frequencies are set symmetrically to the main frequency of the uncontrolled system for the TTMDI-F and I-TTMDI-F and unsymmetrically for the ICTTMDI-F, with larger masses having lower optimum frequencies and damping ratios. Additionally, a closer inspection of Table 4 and Figure 3a reveals that the optimum damping coefficients of both the TMD1 and TMD2 in the I-TTMDI-F and TTMDI-F are simultaneously equal to zero. Amazingly, in these configurations, as ascertained in previous studies [42,43], the damping is localized to the linking damper between the tandem masses. Therefore, there is no damping demand for either the TMD1 or TMD2. Consequently, in a concentrated and straightforward control scheme, a significant amount of input energy is locally dissipated in the linking dashpot in lieu of the TMD1 and TMD2 damping elements.

As a performance criterion, the effective operating frequency bands of the control devices in the frequency domain were evaluated and compared based on Equation (4), where FR(Ω) represents the frequency response at the excitation frequency, Ω. Based on this criterion, a negative value of JF2 demonstrates the effective/efficient performance of control device:

$$JF2\left(\Omega \right)={\displaystyle \frac{{‖H‖}_{2,Controlled}-{‖H‖}_{2,Uncontrolled}}{{‖H‖}_{2,Uncontrolled}}}\times 100$$

(4)

$$JF2\left(\Omega \right)={\displaystyle \frac{{FR(\Omega )}_{Controlled}-{FR(\Omega )}_{Uncontrolled}}{{FR(\Omega )}_{Uncontrolled}}}\times 100$$

(5)

To this end, Figure 4 shows the JF2 variation in the vicinity of the fundamental frequency of the uncontrolled system (3.14 rad/s) for all the control devices. The positive and negative values of the ordinate show the increases and decreases in the frequency response with respect to the uncontrolled system, respectively. Therefore, the pink shaded region (between 2.8 and 3.5 rad/s) indicates the frequency band within which some devices started to show a more efficient performance than that of the uncontrolled structure. Nevertheless, outside this region, all the configurations increased the frequency response with different amplitudes. The effective operating frequency bands of the devices are depicted in this figure with the solid, horizontal, two-sided arrows.

The TMD and ICTTMD-3:4 have the narrowest effective operating frequency bands. To the contrary, all configurations with side inerters have the widest effective bands due to the correlation reduction and the increase in the relative motion between the responses of the two ends of the inerters. Although the TMDI slightly outperformed all the other configurations (an approximately 83% reduction in amplitude) at a frequency of 3.1 rad/s, close to the fundamental frequency of the uncontrolled system, in the effective operating frequency band (pink shaded region), the tandem configurations with side inerters beat the other configurations, with an approximate reduction of almost 80% in amplitude at a frequency of 3.1 rad/s.

4. Seismic Performance Evaluation of Optimum-Designed Tandem-Based Configurations

A comprehensive comparative study was performed among all the presented tandem configurations in the time domain to compare their performances under natural ground motions from past earthquakes. The performance of the controlled SDOF system with a 2% inherent damping ratio was examined under three groups of seven natural earthquakes of FF and NF records with FD and FS characteristics to evaluate the efficiency and robustness of the optimum-designed dampers. To visually illustrate the performances of the proposed control devices, time history comparisons were generated for the uncontrolled and controlled SDOF structure under a selection of earthquake records. Specifically, records 7, 12, 18, and 20 from Table 2—referred to as EQ7, EQ12, EQ18, and EQ20, respectively—were chosen to represent a range of far-fault (FF) and near-fault (NF) ground motions. The selection of these records is further elaborated in Section 4.2. Figure 5 and Figure 6 depict the acceleration and displacement responses for these earthquakes, respectively.

As shown in Figure 5 and Figure 6, the controlled systems exhibited notable reductions in the peak acceleration (approximately 30–40%) and displacement (approximately 30–50%) under far-fault (EQ7) and near-fault forward-directivity (EQ12) motions. This aligns with the frequency domain analysis presented in Figure 4, where the effective operating bandwidths of the control devices overlap significantly with the dominant frequencies of these records. However, the figures also reveal the limited effectiveness of the control devices under near-fault fling-step motions (EQ18, EQ20). This phenomenon arises from the wideband frequency content of these records, which disrupts the frequency alignment critical for the devices’ effectiveness, as previously demonstrated by De Domenico and Ricciardi [71] and Farsijani et al. [72].

Furthermore, three sets of performance criteria were evaluated to compare all the studied tandem-based configurations in the time domain. The first set of performance criteria included four important engineering demand parameters: the maximums and norms of the displacement and absolute acceleration of the main mass [73]. Moreover, the strokes of two tandem masses were also investigated and compared due to the construction limitation concerns. The larger stroke of the TMD demands more accommodation space, which could render its application economically impractical. In all of these criteria, the mean and standard deviation of each criterion were obtained for each device to determine the best configuration under each record set. The second set of performance criteria included the PSDs of the displacement and absolute acceleration of the controlled main mass under four specific earthquake records to scrutinize the relation of the effective operating frequency bands of the control devices and the frequency contents of the earthquakes. The last set of criteria explored different types of structure energies, including elastic strain, kinetic, dissipated, and input energies, under the selected records for the selected controlled and uncontrolled configurations.

4.1. Displacement and Acceleration Response Reductions

The four displacement and acceleration criteria indicate the control performance of each device in comparison with the corresponding values of the least effective control device among all the studied devices to provide a suitable indicator for comparing the control capabilities of the different devices. These are defined according to Equations (6)–(9), where Cd is the control device number: 1, …, 11; q is the earthquake number: 1, …, 21; t is the time: 0 ≤ t ≤ Tq; d and a are the displacement and absolute acceleration of the main mass, respectively; and $\left|.\right|$ and $‖.‖$ express the absolute value and norm of the responses, respectively:

$$JT1\left(Cd,q\right)={\displaystyle \frac{\underset{t}{\max }\left|d_{Cd}\left(t,q\right)\right|-\underset{t,\mathit{Cd}}{\max }\left|d_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|d_{Cd}\left(t,q\right)\right|}}\times 100=({\displaystyle \frac{\underset{t}{\max }\left|d_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|d_{Cd}\left(t,q\right)\right|}}-1)\times 100$$

(6)

$$JT2\left(Cd,q\right)={\displaystyle \frac{‖d_{Cd}\left(t,q\right)‖-\underset{\mathit{Cd}}{\max }‖d_{Cd}\left(t,q\right)‖}{\underset{\mathit{Cd}}{\max }‖d_{Cd}\left(t,q\right)‖}}\times 100=({\displaystyle \frac{‖d_{Cd}\left(t,q\right)‖}{\underset{\mathit{Cd}}{\max }‖d_{Cd}\left(t,q\right)‖}}-1)\times 100$$

(7)

$$JT3\left(Cd,q\right)={\displaystyle \frac{\underset{t}{\max }\left|a_{Cd}\left(t,q\right)\right|-\underset{t,\mathit{Cd}}{\max }\left|a_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|a_{Cd}\left(t,q\right)\right|}}\times 100=({\displaystyle \frac{\underset{t}{\max }\left|a_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|a_{Cd}\left(t,q\right)\right|}}-1)\times 100$$

(8)

$$JT4\left(Cd,q\right)={\displaystyle \frac{‖a_{Cd}\left(t,q\right)‖-\underset{\mathit{Cd}}{\max }‖a_{Cd}\left(t,q\right)‖}{\underset{\mathit{cd}}{\max }‖a_{Cd}\left(t,q\right)‖}}\times 100=({\displaystyle \frac{‖a_{Cd}\left(t,q\right)‖}{\underset{\mathit{cd}}{\max }‖a_{Cd}\left(t,q\right)‖}}-1)\times 100$$

(9)

The values of JT1–JT4 in each record are presented in Figure 7, Figure 8, Figure 9 and Figure 10, respectively. In each figure, three a-c plots are provided to facilitate comparing the performances of the different devices for a specific criterion under FF records, NF records with FD, and NF records with FS, respectively. Additionally, at the right end of each plot, the means and standard deviations of the performances of the individual devices are depicted under the corresponding record set.

Figure 7 and Figure 8 show the reductions in the maximum and norm of the displacement of the main mass, respectively, for each device under all three record sets. All devices exhibited completely different behaviors under different records in the same group, emphasizing the dependency of the performance on the characteristics of the record, and particularly its frequency content. Figure 7 shows that the maximum reductions in the JT1 (maximum displacement) were about 38% using the TTMDI (for free and predefined mass ratios), 33% using the TTMDI (for free and predefined mass ratios), and 38% using the ICTTMDI-F under FF records, NF records with FD, and NF records with FS, respectively. It should be noted that under the FF and NF with FD records, the ICTTMDI-F shows close performances of 35% and 31% reductions in the JT1, respectively. When comparing the mean of the performance for each device under different records, however, the reduction in the JT1 has a more uniform pattern. Therefore, the control devices can be divided into three categories based on their performances in the reduction in the mean of the maximum displacements. These categories are the TMD-based devices (i.e., the TMD, TTMD, and ICTTMD), the I-TTMDI-F, and all the rest of the TMDI-based devices, which show under 3%, between 10 and 15%, and between 15 and 20% reductions in the response, respectively. Interestingly, the TTMDI configurations with free or preselected mass distributions, as well as the ICTTMDI-F, outperformed all the other configurations for all the earthquake records in terms of reducing the mean of the maximum displacements. However, the comparison of the standard deviations shows that the increase in the performance is simultaneous with the rise in the standard deviation and, therefore, the reduction in robustness. Remarkably, the standard deviations of the TMD-based configurations increased considerably under the NF records with FD due to the wideband frequency contents of the records and narrow effective operating frequency bands of the devices. To the contrary, the standard deviations for the TMDI-based devices remained almost unchanged within all the different record groups. These observations are in good alignment with the results from previous studies by Kaveh et al. [51] and Pietrosanti et al. [52].

The results of the reductions in the norm of the displacement (JT2) are presented in Figure 8, where a similar trend for reducing the norm of the displacement is observed, with the TMDI-based devices showing consistently superior performances; however, the devices generally exhibited enhanced performances (between 5 and 15%) in comparison with those for the JT1, indicating their better effectiveness at controlling the overall vibrations rather than just the peak responses. These observations align with the findings from previous studies on TMDI-based devices by Pietrosanti et al. [52], which showed that incorporating the inertance provides more effective displacement mitigation under various seismic records. Moreover, a similar categorization can be suggested for the TMD-based control devices (i.e., the TMD, TTMD, and ICTTMD), the I-TTMDI, and all the rest of the TMDI-based devices. Although the performances of all the categories increased in reducing the JT2 in comparison with the JT1, the TMD-based devices show mean values closer to zero with larger standard deviations, demonstrating that their performances remained almost unchanged under NF records with FD and FS. This pattern stems from the inherent limitation of the TMD-based devices when addressing the wideband frequency contents of these records.

Considering the absolute accelerations of the main mass, Figure 9 and Figure 10 depict the reductions in the maximum and norm of the main mass absolute acceleration, respectively, for each device under all three record sets. All devices exhibited utterly different behaviors under different records in a group, indicating that all the instruments depended on the specifications of the record and, in particular, its frequency content, as previously highlighted in studies by De Domenico and Ricciardi [71] and Dai et al. [74]. In Figure 9, the maximum reduction in the JT3 was about 31% (using either the TTMDI-F or I-TTMDI-F) under all record sets. A comparison of the means of the performances for the individual devices under different record sets demonstrates that the maximum reduction in the JT3 has a more uniform pattern. Therefore, the control devices can be divided into two categories based on their performances in the reduction in the means of maximum acceleration. These categories are TMD-based control devices and TMDI-based control devices, and their related reductions are under 3% and about 15%, respectively. Furthermore, the standard deviations show a more uniform pattern for the maximum acceleration reductions, with small and large standard deviations for the TMD-based and TMDI-based configurations, respectively. In Figure 10, the maximum reductions in the JT4, the norm of absolute acceleration, were approximately in the range of 35–43% (using either the TTMDI or I-TTMDI) under all record sets. Additionally, the reductions in the norm of absolute acceleration (JT4) for the TMD-based and TMDI-based devices are similar to the reductions in the norm of displacement (JT2).

Figure 7, Figure 8, Figure 9 and Figure 10 show that the control devices performed differently under different earthquake records; e.g., most devices had excellent performances under earthquake record 12 (EQ12), while the performances of the same devices under EQ18 were unsatisfactory. As mentioned previously, these different and sometimes contradictory performances are usually attributed to the interaction of the record frequency content and the system frequency response, which is a function of the modification of the dynamic characteristics of the controlled system. Therefore, before evaluating the strokes of different control devices, in the next section, the PSDs of the earthquake records, as well as the PSDs of the displacement and acceleration of the main mass under selected records, are compared to give some insight into the different and contradictory behaviors of the control devices under different ground motions.

4.2. PSD Comparison

The PSD of a signal shows the density of the energy in the frequency content of the signal. Consequently, the PSD of structural responses could be representative of the energy density of the input signal and the effective operating frequency band of the structure simultaneously. Therefore, in this section, four earthquake (EQ) records, shown in Table 2, have been selected to further elaborate the performances of the control devices and simultaneously evaluate the relationship between the effective operating frequency bands of the control devices and the frequency contents of the records. The selected record numbers are 7, 12, 18, and 20, and they were selected such that at least each set had a representative in this comparison. Hereafter, these records are referred to as EQ7, EQ12, EQ18, and EQ20, respectively. These records were selected among all the other records due to the distinguished performances of all the control devices based on the reductions in the maximum and norm of the main mass displacement and absolute acceleration under them. Under EQ7, the results (Figure 7, Figure 8, Figure 9 and Figure 10) illustrate the outstanding performances of all the devices in decreasing the norms of the responses, while the reductions in the maximum responses were insignificant. In the case of EQ12, all devices exhibited promising performances in reducing the maximums and norms of the responses simultaneously, whereas considering EQ18 and EQ20, all devices showed the lowest control efficiencies in reducing all the performance criteria. The PSDs of these records and the PSD values of the displacements and absolute accelerations of the main mass for the uncontrolled, TMD-based control, and TMDI-based control devices are illustrated in Figure 11 and Figure 12, with both axes in logarithmic scale. The classical TMD and TMDI represent the TMD-based and TMDI-based devices in these figures. To this end, Figure 11 presents the PSDs under EQ7 and EQ12, i.e., the earthquakes under which all devices reduced at least one criterion remarkably, while Figure 12 shows the PSDs under EQ18 and EQ20, which are records under which all devices failed to significantly reduce the indicators. In both figures, the abscissa shows the frequency, and the area adjacent to the fundamental frequency of the uncontrolled structure between 0.1 and 1 Hz is highlighted, where the energy density of the earthquakes may have a profound effect on the responses of the structures.

Figure 11a,b illustrate the PSD values of EQ7 and EQ12, respectively, where the smoothened form that is calculated by the moving average is also presented in red for further comparison purposes. Similarly, Figure 12a,b show the PSD values of EQ18 and EQ20 and their smoothened curves, respectively. The region of structural importance (shaded area) shows that by decreasing the frequency, the PSD values of EQ7 and EQ12 decrease with a significant slope, while EQ18 and EQ20 do not exhibit such trends. For the latter two records, the PSD changes were much less sharp, and even the PSD of EQ20 remained almost constant with the decreasing frequency. In the second (part c and d) and third (parts e and f) rows of Figure 11 and Figure 12, the PSD values of the displacement and acceleration of the main mass under the corresponding records are presented, respectively.

In the second and third rows of Figure 11, the representative devices, i.e., the TMD and TMDI, not only show quite different behaviors in the maximum peak reduction, but they also both performed quite well in the pre-peak region at lower frequencies. In both the peak and pre-peak areas, although the TMD-based devices were clearly effective at reducing the response, the TMDI-based devices were more effective at reducing both the displacement and acceleration of the main mass, consistent with the reported work from previous studies by Pietrosanti et al. [52] and Marian and Giaralis [61].

To the contrary, in Figure 12, there is no significant difference in the performances of the different control devices in the reduction in either the peak or pre-peak areas. Since both the TMD and TMDI are tuned to the natural frequency of the main mass, in the controlled structures, anti-resonance is observed in the fundamental period of the main mass in both Figure 11 and Figure 12. Due to the locations of the two new resonance frequencies in the PSDs of the controlled structures and the variation patterns of the PSDs of the earthquakes, the PSDs of the responses in these new resonances decrease and increase in Figure 11 and Figure 12, respectively. These findings align with the results reported by Cao and Li [43], demonstrating similar resonance trends under white-noise base excitations. Since in these figures, only the performances of the TMD and TMDI (as the representatives of different TMD-based and TMDI-based devices) are investigated, Figure 13 is presented to further investigate the performances of the individual control schemes and to take a closer look at the effect of the interaction between the spectral energy distributions of the earthquake records and the effective operating frequency bands of the control devices on their seismic performances.

In Figure 13a, part of the PSD of the records is magnified in the range of 2–4.5 rad/s to investigate the influence of the effective operating frequency bands of the control devices on their seismic performances and its relationship with the earthquake frequency content. Both the abscissa and ordinates have linear scales to demonstrate a clear distribution of the earthquake energy in different frequencies. In Figure 13b, for the convenience of the readers, Figure 4 is re-presented to allow for the comparison and investigation of the interaction between the frequency contents of the earthquake records and the effective operating frequency bands of the control devices. In Figure 13a, it is clear that the earthquake energy remained unchanged or dropped slightly outside the effective operating frequency band under EQ7 and EQ12, while the earthquake energy increased outside this range under EQ18 and EQ20. Therefore, none of the control devices successfully reduced the structural responses under EQ18 and EQ20, while they performed well under EQ7 and EQ12.

4.3. Assessment of Strokes

From the first application of TMD devices in buildings, the large strokes of TMDs have been a practical limitation in implementing them. Aiming at comparing the strokes of all the tandem-based devices with those of the classical TMD, the JT5 and JT6 performance criteria (Equations (10) and (11)) were defined for each tandem mass (i.e., TMD1 and TMD2, respectively), in which the strokes were normalized to the corresponding values of the device with the largest stroke. In Equations (10) and (11), S1 and S2 denote the strokes of TMD1 and TMD2 in the tandem configurations, respectively:

$$JT5\left(Cd,q\right)={\displaystyle \frac{\underset{t}{\max }\left|{S1}_{Cd}\left(t,q\right)\right|-\underset{t,\mathit{Cd}}{\max }\left|{S1}_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|{S1}_{Cd}\left(t,q\right)\right|}}$$

(10)

$$JT6\left(Cd,q\right)={\displaystyle \frac{\underset{t}{\max }\left|{S2}_{Cd}\left(t,q\right)\right|-\underset{t,\mathit{Cd}}{\max }\left|{S2}_{Cd}\left(t,q\right)\right|}{\underset{t,\mathit{Cd}}{\max }\left|{S2}_{Cd}\left(t,q\right)\right|}}$$

(11)

Figure 14 and Figure 15 demonstrate the variation trends of the nondimensional strokes of the TMD1 and TMD2, respectively. Both figures show that the TTMD-1:1 and TTMD-3:4 exhibit the largest strokes, while the largest mean reductions (between 60 and 70%) in both the JT5 and JT6 are observed in the TMDI under all record sets. In general, the TMDI-based devices acted significantly differently from the TMD-based ones, exhibiting superior performances to those of the TMD-based devices due to the merits of the inerters. Moreover, among the TMDI-based devices, the TMDI outperformed all the other devices. This can be attributed to the fact that the TMDI has just one mass that is larger than that of any of the two mass blocks in the tandem-based configurations and therefore needs a higher viscous damping value, which, in turn, limits its stroke. In general, the higher the actual or apparent mass, the higher the optimum damping and, consequently, a smaller stroke of the device is obtained. However, it should be noted that the lower requirement of damping could be beneficial in implementing tandem-based devices.

To provide a concise overview of the numerical performances of the different control device configurations studied,

Table 5. Summary table of control device performances.

Control Device	H₂ Norm (JF1)	JT1 (Max Disp. ↓%)	JT2 (Norm Disp. ↓%)	JT3 (Max Accel. ↓%)	JT4 (Norm Accel. ↓%)	JT5/JT6 (Stroke ↓%)	Notes
TMD	0.7045	<3%	<5%	<3%	<4%	20–25%	Weak performance overall; high stroke demand
TMDI	0.5133	15–20%	20%	15%	20–25%	60–70%	Excellent stroke performance; robust across metrics
TTMD	0.6782–0.6814	10–12%	13%	10%	12%	10–20%	Better than TMD but with high strokes; limited robustness
TTMDI	0.4889–0.4898	18–20%	22–24%	20–22%	25%	45–50%	Balanced performance across all metrics
ICTTMDI-F	0.5033	19%	23%	21%	26%	50–55%	Close to TTMDI; competitive overall
I-TTMDI-F	0.4834	20%	24%	22%	27%	55–60%	Best performer across all indices and strokes

summarizes their effectiveness based on six key evaluation indices (JT1–JT6) and the H₂ norm (JF1). These values represent the average reductions observed across 21 earthquake records, as detailed in the preceding performance analyses. The table highlights that the I-TTMDI-F configuration exhibited the best performance across all the metrics, while also noting the limited effectiveness and higher stroke demands of the TMD-based devices.

4.4. Structure Energy

In this section, the performance and effectiveness of the control devices are further investigated by comparing the different energies in the structure under the four previously selected records. To this end, the kinetic energy (KE), damping energy (DE), elastic strain energy (SE), and input energy (IE), defined as Equations (12)–(15), respectively, are further investigated:

$$KE={\int }_{t=0}^{t=t_k}{\ddot{Y}}^{T}MdY={\displaystyle \frac{1}{2}}{\dot{Y}}^T\left(t_k\right)M\dot{Y}\left(t_k\right)$$

(12)

$$DE={\int }_{t=0}^{t=t_k}{\dot{X}}^{T}CdX={\int }_{t=0}^{t=t_k}{\dot{X}}^{T}C\dot{X}dt$$

(13)

$$SE={\int }_{t=0}^{t=t_k}{X}^{T}KdX={\displaystyle \frac{1}{2}}{X}^T\left(t_k\right)KX\left(t_k\right)$$

(14)

$$IE={\int }_{t=0}^{t=t_k}{\ddot{Y}}^{T}Md{X}_g={\int }_{t=0}^{t=t_k}{\ddot{Y}}^{T}M{\dot{X}}_{g}dt$$

(15)

The time histories of these energies were calculated and compared for uncontrolled and controlled structures with the TMD, TMDI, and newly proposed ICTTMDI-F and I-TTMDI-F under EQ7, EQ12, EQ18, and EQ20, shown in Figure 16, Figure 17, Figure 18 and Figure 19, respectively. In all the figures, the plots of (a) through (d) show the SE, KE, DE, and IE, respectively. It should be noted that there are significant differences between the elastic and kinetic energies of the uncontrolled, TMD-based, and TMDI-based controlled structures. By increasing the effective operating frequency bands of the control devices, these energies decreased during the whole durations of the records for the cases of EQ7 and EQ12, as shown in Figure 16 and Figure 17, respectively. However, none of the control devices showed peak reduction efficiencies under EQ18 and EQ20. Even so, at most points in the post-peak part, the time histories for these devices showed greater responses than those of the uncontrolled system, as shown in Figure 18 and Figure 19, respectively.

This observation is justifiable based on the results presented in Section 4.2 on the energy density records on the fundamental frequencies of the uncontrolled system and the effective operating frequency bands of the various devices, as previously discussed by Pietrosanti et al. [52]. The interaction between the frequency contents of the earthquakes and the effective operating frequency bands of the devices resulted in desirable performances of all the devices under EQ7 and EQ12 and undesirable performances of all the devices under EQ18 and EQ20. Additionally, under EQ7 and EQ12, by increasing the effective operating frequency band of the device, significant differences between the classical devices and the newly proposed devices (ICTTMDI-F and I-TTMDI-F) were observed, of which the latter showed better performances due to their large effective operating frequency bands. It can be concluded that since the energy density records are not under the designers’ control, all efforts should be concentrated on employing devices with larger bandwidths, following the recommendations of Ma et al. [46]. Therefore, the newly proposed devices (ICTTMDI-F and I-TTMDI-F) would probably have the best possible performances among all the studied devices under identical records and guarantee robustness due to their larger bandwidths [53].

Similar results can be deduced more clearly from the DE and IE energies (the plots of (c) and (d), respectively). The amounts of these energies for the various control devices under EQ7 and EQ12 are different from those under EQ18 and EQ20. Accordingly, it can be observed that, generally, under EQ7 and EQ12, the uncontrolled and I-TTMDI-F controlled structures had the largest and smallest energies, respectively, during the entire duration of the records, and particularly at the end section of the records. Similar results were observed in the study conducted by Zhao et al. [75], which highlighted the intrinsic ability of inerter-based tuned mass dampers to effectively reduce the input energy. To the contrary, opposite trends were observed in the cases of EQ18 and EQ20. This could be justified based on Figure 12, where a device with a larger effective operating frequency band shows the worst performance outside its bandwidth. Therefore, they even could exacerbate the responses if the energies of the records remain unchanged or increase outside their effective operating frequency bands. Similar results for the other records can be presented but are omitted herein for the sake of brevity.

5. Summary and Conclusions

In the current work, two new tandem-based tuned mass damper inerters are introduced, inspired by recently proposed tandem-based configurations. These two new devices extend and integrate the recently proposed tandem-based arrangements in the literature to fully exploit the different merits of the previously proposed configurations. A uniform framework was established to optimize and compare the performances of six different optimum-designed tandem-based control devices. In this framework, the performances of the different tandem-based TMDs for the structural control of an SDOF structure were compared in both the frequency and time domains under a suitable number of natural earthquake records with different characteristics. Three different types of performance criteria, i.e., the maximum and norm of the displacement and absolute acceleration of the main mass, the strokes of the TMDs, the PSDs of the absolute acceleration and displacement of the main mass, and the structure’s energies, were evaluated under 21 FF and NF records with FD and FS. This study followed a clear and structured process to design, optimize, and evaluate the proposed configurations, which is outlined in the points below:

• In the optimization process of the newly proposed devices, the only preselected design parameters were the total mass ratio (TMR) and total inertance ratio (TIR), and all other design parameters, particularly the distributions of the mass and inertance, were assumed to be the design variables. The optimally designed dampers using the norm of the displacement of the main mass not only revealed the better performances of the tandem-based configurations but also indicated an asymmetric distribution of the TMR, a symmetric distribution of the TIR to the side inerters, and the redundancy of the connecting inerter between the two masses due to the mass amplification and negative stiffness effects. Furthermore, similar to previous studies, the damping was found to be localized to the linking damper between the tandem masses. Therefore, there is no damping demand for either the TMD1 or TMD2. Remarkably, an equivalent relative mass ratio (RMR = ${\displaystyle \frac{m_{e1}}{m_{e2}}}$) equal to about 0.7 was obtained for all the newly proposed tandem-based control devices, which is close to the previously reported RMR for the optimum TTMD in the literature;
• The results of the frequency domain analysis showed the outstanding performances of the newly proposed devices, i.e., the I-TTMDI-F, ICTTMDI-F, and TTMDI-F, in reducing both the peak and norm responses of the uncontrolled system and having the largest effective operating frequency bands. However, the worst performance in the frequency domain in reducing the maximum and norm values of the response was found for the TMD-based devices, especially the ICTTMD, for which all the inertance was assigned to the connected inerter between the two tandem masses. This not only did not improve the control performance of the device but also reduced the performance compared to the other devices without any grounded inerter;
• Finally, the performances of all the control devices were evaluated under far-fault (FF) and near-fault (NF) records with forward-directivity (FD) and fling-step (FS) characteristics in terms of the different performance criteria. Although the results of the evaluation of the first group of performance criteria demonstrated scattered results under each record, similar to the frequency domain results, they emphasized the importance of the existence of side inerters and the redundancy of the connecting inerter between the tandem masses in reducing all the performance criteria (i.e., the maximum and norm of the main mass displacement and absolute acceleration). Despite the poor performance of the ICTTMD due to the negative stiffness effect of the connecting inerter, it was able to reduce the strokes by 20 to 30% compared to the other TMD-based devices without significantly reducing the performance; this can be considered an important practical measure for roof-top devices when space is a practical challenge. Additionally, the evaluation of the power spectral density (PSD) and energy in the structure justifies the performances of the devices under each record by the interaction between the effective operating frequency bands of the devices and the frequency contents of the records. Moreover, by considering the effective operating frequency band of each device, the worst performance was outside its effective operating frequency band, and the spectrum distribution of the earthquake energy dictates the amount of energy experienced by the structure. Hence, the importance of devices with large frequency bandwidths was highlighted.

6. Practical Considerations and Recommendations for Future Studies

The findings of this study offer multiple points of direct application for engineering practice, particularly in seismic design and retrofitting. The comparative results across six performance metrics (JT1–JT6) and the H₂ norm (JF1) allow engineers to identify which damper configurations best meet the desired performance objectives. For instance, the I-TTMDI-F consistently outperformed the other configurations across the displacement, acceleration, and stroke metrics. Furthermore, the results of the current study provide optimized damping ratios, frequency ratios, and mass/inertance distributions for each configuration. These values—developed under realistic conditions (2% structural damping, 1% total mass, 5% inertance)—can serve as reference input for actual design or early-phase feasibility studies. In addition, the results show that the stroke demands (JT5/JT6) were reduced by up to 60–70% in the TMDI-based configurations compared to standard TMDs. This is crucial for planning installation space, especially for constrained mechanical floors or retrofitting applications. Given the compactness implied by the stroke reduction and the passive nature of these systems, engineers can use the provided data to assess the practicality of implementing TTMDI devices in existing buildings—particularly where conventional TMDs are infeasible due to size constraints. Finally, the robustness of the results—derived from 21 ground motions including both the near- and far-fault types—supports the use of advanced TTMDI configurations in performance-based seismic design.

Further research is needed to validate the ICTTMDI and I-TTMDI configurations with other advanced vibration control systems, like friction pendulum isolators, particularly in MDOF primary structures. Additionally, while the current study focused on linear structural behavior, future work should investigate the damper performance under nonlinear structural dynamics, including post-yield scenarios wherein the stiffness and damping properties evolve. The observed performance degradation outside the operating bandwidth requires further research to develop potential solutions, such as adaptive tuning mechanisms or semi-active control elements. Furthermore, experimental investigations are essential to validate the results provided in this work, ensuring their applicability. This will facilitate the wider implementation of tandem-based vibration control systems in the design of buildings for seismic resilience.