Kawamata first introduced a liquid mass pump in 1973 that leverages fluid inertia to reduce seismic responses in structures. This is viewed as the origin of the inertial component. Inerters were first named by Smith in 2002, which are two-terminal devices that can significantly amplify physical mass by converting translational motion to rotational motion. It was not until Saito et al. used inertial elements in 2002 to enhance mass and damping of viscous damping devices that the mass amplification effect of inertial elements was recognized. Since then, the combination of inertial devices and traditional passive vibration control devices has received more and more attention.

Ikago [1] proposed a method of combining an inertial mass in 2012 with a viscous mass damper to create a tuned viscous mass damper (TVMD), which has shown better control efficiency than traditional viscous mass dampers (VMDs) and viscous dampers (VDs). Based on this, in 2013 Garrido replaced the conventional viscous damper (VD) with a tuned viscous mass damper (TVMD) in a tuned mass damper (TMD) to create a rotating inertia dual tuned mass damper (RIDTMD) [10,12]. Marian et al. [16] developed a tuned mass damper inertial (TMDI) device in 2014 by parallel connecting an inertial mass with the damper and spring components of a TMD. Ruiz et al. [26,27] introduced the concept of an inertiabased dynamic vibration absorber (IDVA) in 2015, which replaced the damper element in a TMD with different configurations of an inertial mass, damper, and spring connected in series or parallel. Cao et al. [24] divided a TMDI into two serially connected TMDIs in 2019 to create a tuned serially connected TMDI system (TTMDIs). This was followed by Luo et al. in 2016, Cao et al. in 2022 for inertia-based passive control devices for foundation isolation, and Giaralis et al. in 2017, and Petrini et al. in 2020 for seismic and windinduced vibration control in high-rise buildings [28]. Creatively, Brzeski et al. [29] created the tuned inertial damper (TID) in 2014 by replacing the mass element in the TMD with an inertial element. TMDI is derived from TMD, which was first proposed by Frahm for reducing the multi-degree-of-freedom (SDOF) response of a primary system with a tuned linear spring to harmonic forces within a narrow frequency [30–34].

The concept of an ideal inertial device was first proposed by Smith in 2002, and in 2004 Papageorgiou and Smith experimentally demonstrated that by incorporating a rack and pinion or ball screw mechanism, the translational energy of the structure could be converted into rotational energy stored in a relatively light rotating disc [12]. Experiments by Wang et al. in 2011 and Swift et al. [17] in 2013 verified the implementation of hydraulic inertial devices with inertia values b almost independent of the mass of the physical device. The ideal inertial device can be understood as an inertial amplifying device since it acts as a “weightless” mass b when “grounded” at either of its terminals. Prompted by this consideration, tuned mass dampers-inertial systems-have been proposed which enhance the vibration control capabilities of classical tuned mass dampers. Therefore, Marian et al. [32] have demonstrated in 2013 and 2014 that utilizing the inertial amplification property can significantly improve the vibration control ability of a device under the same additional mass (thus weight) conditions (see also Fig. 2) [15].

In 2013 and 2014, Marian et al. [13,32,35] proposed a generalization of the classical Tuned Mass Damper (TMD), which includes an “inertial” device called the Tuned Mass Damper with an Inertia device (TMDI). The TMDI introduces inertia into the conventional vibration absorber, leading to the development of various inertia-based dampers. The study of TMDI plays an important role in engineering vibration control, particularly in high-rise buildings or bridges located in areas susceptible to strong winds or earthquakes. The TMDI is an extension of the classical linear TMD, which benefits from the mass amplification effect of the inertia device, or the so-called “inertial amplification” to enhance the vibration suppression capability of the TMD [32–36]. Zhang et al. [37] investigated the structural control of high-rise buildings under wind and earthquake loads and verified that under a multi-modal control mechanism concerning TMDI, it is mainly determined by the contribution of two aspects of TMDI: the damping effect associated with inertia, auxiliary mass, and damping coefficient and the negative stiffness effect associated with auxiliary mass. In the vibration reduction of tall buildings with wind excitations, the performance of the TMDI is heavily influenced by the floor to which the inerter is connected. Therefore, the vibration reduction performance of the building is calculated as a function of this parameter. It is important for designers to carefully consider the installation location of the inerter as it can significantly impact the overall effectiveness of the TMDI system. Optimal TMDI performance is achieved when the inerter end connected to the ground floor experiences zero absolute acceleration. However, connecting the inerter to a floor located beneath the pendulum mass (see Fig. 3) means that TMDI can only surpass TMD if said inerter is connected to a floor at around one-third of the building’s height (see Fig. 3) [38]. In the most realistic scenario, TMDI exhibits significantly poorer performance compared to the classical TMD when the inerter is linked to the mass located close to the ground [38,39].

The study investigates two TMDI topologies [40]. The traditional topology involves connecting the inerter in parallel with the spring and viscous damper. The nonlinear electromagnetic resonant shunt tuned mass damper-inerters (NERS-TMDI) has two different topologies. One topology involves connecting the spring, inertial damper, and viscous damper in series to increase the inertia force of the TMDI. The other topology connects these components in parallel to the stiffness of the TMDI’s mass-spring oscillator and aims to produce an additional degree of freedom. The first topology aims to increase the inertia force of TMDI, while the second aims to produce an additional degree of freedom. Results show that the TMDI with the spring, inerter, and viscous damper in series results in an increased degree of freedom of the damper while avoiding significant friction in the inerter. This benefit is realized through the series connection of the spring, inerter, and viscous damper. The use of different TMDs and inerters in base-isolated buildings results in completely different dynamic performance of the structure. Therefore, de Domenico et al. [25] proposed six different strategies for a systematic comparative study, with each strategy characterized by a specific combination of mass-spring-damper elements arranged in series or parallel to alleviate the displacement of base-isolated structures. The specific schematic is shown in Fig. 4 [41]:

With reference to previous indepth studies on vibration isolators, we have a preliminary understanding of the development of TMDI in vibration control [42]. As shown in Fig. 5 [43], Classic TMD coupled with an inerter in different TMDI topologies are proposed to suppress excessive wind-induced vibrations that cause discomfort to residents in high-rise buildings [43]. At the same time, it has been proposed to use two different structures for the Nonlinear Electromagnetic Resonant Series-Tuned Mass Damper with Inerter (NERS-TMDI) [44,45]. Results show that NERS-TMDI is superior to existing Nonlinear Tuned Mass Dampers (NTMD) and Electromagnetic Resonant Series-Tuned Mass Damper with Inerter (ERS-TMDI) in terms of vibration control. It has been found that the first configuration of NERS-TMDI consistently performs better than the second configuration in reducing vibration and harvesting energy. The vibration behavior of ERS-TMDI system is derived when subjected to wind-induced excitation on the primary structure (see Fig. 5) [5]. Simultaneous, the performance of the ERS-TMDI system in vibration reduction and energy harvesting is numerically analyzed. At a high inertial ratio, research has shown that the vibration reduction performance of the primary structure deformation and relative deformation of ERS-TMDI is better. ERS-TMDI reduces the resonance peak by 35% compared to the classic TMD, and by 18% compared to ERS-TMDs, significantly improving the vibration reduction effect. Finally, Luo et al. [5,22,46] introduced the combination of ERS-TMDI and single-degree-of-freedom systems, and the results showed that ERS-TMDI is superior to classic TMD, ERS-TMDs, and series-connected double-mass TMD systems in terms of structural damage protection.

Based on the differences between internets devices, the following Table 1 shows the available vibration control devices and their characteristics.

Since Liu et al. [53] derived the motion differential equation of the Tuned Frequency Mass Damper (TFMD) under harmonic excitation. The frequency domain displacement response of the transmission conductor acting on TMDI was analyzed using the Fourier transform. A collision-based optimization (CBO) algorithm based on heuristic algorithm was used to optimize the response of the transmission elevated-TMDI structure. Based on the CBO optimization, the transmission line TMDI system’s parameters were optimized to minimize its displacement. Through his research, he has shown that by optimizing TMDI parameters using CBO, the displacement response of the transmission line can be effectively controlled, leading to a significant improvement in vibration control efficiency. Salvi et al. introduced a new structure that simultaneously suppresses vibration and collects energy in a harmonic excitation structure based on a damped SDOF dynamic system (see Fig. 6) [51]. The stiffness and damping characteristics of TMD can be adjusted using a simple design formula to achieve enhanced vibration suppression (based on the peak displacement of the SDOF primary structure) and energy harvesting (based on the relative velocity at the EM terminal). This can be done at a similar resonance frequency with a fixed mass attached to TMD, as illustrated in Fig. 6. Based on this situation, there exists a balance between vibration control and energy harvesting. Therefore, energy harvesting-enabled tuned mass-damper-inerter (EH-TMDI) is inspiring for future multi-objective optimization design processes. For the SDOF primary structure, the TMDI includes an inertial device connecting the additional mass of the classical TMD to the ground, as shown in Fig. 6h. The mass of the classical TMD is connected to a skyhook configuration on the ground as shown in Fig. 6b where β is the inertia ratio and µ is the mass ratio [51].

Due to the excellent mechanical performance of TMDI, TMDI has become an exciting candidate device in the field of passive vibration control of space-constrained offshore wind turbines. Fitzgerald et al. [54] have shown that TMDIs offer significant damping potential for floating offshore wind turbine (FOWT) towers. For the same damper mass, this new type of damper performs significantly better than a conventional TMD, and the TMDIs can significantly improve the reliability of FOWT towers in adverse weather conditions with wind and wave dislocation. The application of a passive TMDI in the vibration control of a mast-style FOWT tower was studied, and the optimal tuning parameters of the simplified fixed-base tower under white noise excitation were derived [47]. The optimal TMDI design parameters were derived as a function of TMD mass and inerter constant b to minimize the relative displacement variance of the undamped SDOF structure subjected to white noise excitation [55]. The findings indicate that, with a fixed TMD mass, the optimized TMDI setup is superior to the traditional TMD configuration in mitigating the displacement variance of an undamped SDOF structure when it undergoes white noise excitation. In particular, the hybrid control of the structure based on TMDI and the base-isolation subsystem under seismic excitation was studied (see Fig. 7), and TMDI was optimized by implementing a simplified method based on minimizing the variance of the base isolation subsystem’s displacement (see Fig. 7) [56]. To increase the safety and availability of a beam structure while controlling its vibration, it has been suggested to use Lagrange’s equation in conjunction with a smart monitoring sensor for electrical grids. This approach would aid in the calculation of the response of the beam structure to seismic acceleration excitation and the corresponding displacement values, and the displacement response formula for the beam structure under kinetic force are derived [56].

(1)[M]{y¨}+[C]{y˙}+[K]{y}=−[M][E]x¨g(t)+FTFT=Dd[cd(x˙d−y˙)+kd(xd−y)](b+md)x¨d+cd(xd−y˙)+kd(xd−y)=−mdx¨g(t)

where the displacement of the TMDI with respect to the base is denoted as x_d (t). Let m_d, b, c_d and k_d be the mass, inertia, damping and stiffness of the TMDI, respectively.

The effect of the position of the TMDI in controlling wind-induced vibration of a flexible structure is investigated as shown in Fig. 8, and the equations of motion for a TMDI structure system considering the inertial position are developed below [22,23,57,58]. Meanwhile, based on the mass amplification effect, and control robustness conditions, the effect of inertial position on the control effectiveness, optimization of design parameters, and high-mode damping effect of TMDI is discussed [57].

[M+dH2H2T−dH2−dH2Tmt+d]{X¨x¨t}+[C+ctH1H1T−ctH1−ctH1Tct]{X˙x˙t}

(2)+[K+ktH1H1T−ktH1−ktH1Tkt]{Xxt}={F0}

where H₁ and H₂ are the position vectors of the damping and inertial elements; X¨, X˙ and X are the acceleration, velocity, and displacement response vectors of the primary structure; M, C and K are the mass matrix, damping matrix and stiffness matrix of the primary structure.

Based on the adverse effects of crosswind vortex-induced resonance on high-rise chimneys, high-rise chimneys can be reduced to generalised SDOF structures. Based on the simulated filtered wind load spectra, analytical solutions were derived for the wind-induced vibration response of the structure with the TMDI installed. Subsequently, a parameterized analysis of the optimized design parameters of TMDI is conducted. From the basic motion equations, an analytical expression for the efficiency of TMDI wind-induced vibration response control for the generalized SDOF high-rise chimney structure is derived based on the filtered representation method. Through parameter analysis, empirical formulas for TMDI’s optimal parameters, optimal control ratio, and effective TMD mass ratio are obtained [59–61].

In the vibration control of high-rise buildings, TMDI can be used to control the structure of high-rise buildings under wind and earthquake loads through the inherent multi-mode control effectiveness of TMDI. The analysis and research of TMDI’s multi-mode control effects suggest that the damping effect of the weighted difference of mode coordinates between two ends of TMDI is the main reason for the multi-mode control effectiveness of TMDI. Finally, the change trend of the equivalent damping ratio of TMDI-controlled structures under any mode is analyzed through the established model, and the design parameters that have the greatest impact on the multi-mode control effectiveness of TMDI are determined. In addition, when inertial dampers are used to reduce the wind-induced response of high-rise buildings, Studies have shown that optimally designed TMDIs or TIDs are better at suppressing wind-induced vibrations than TMDs, especially in terms of acceleration response, when assigned low or zero additional mass. Based on the concept of structural dynamics, the motion equations of the MDOF structure controlled by TMDI can be expressed in a matrix-vector [13]:

(3)Mx¨(t)+Cx˙(t)+Kx(t)=p(t)+1iFi(t)−1aFa(t)

where M, C, and K are the mass matrix, viscous damping matrix, and stiffness matrix of the primary structure, respectively. x˙ and x¨ denote the corresponding velocity and acceleration response vectors. p(t) denotes the excitation vector acting on each degree of freedom at a moment.

The generalized SDOF system is only suitable for the design of TMDS and inter-floor TMDIs adjusted to the basic vibration mode. For TMD-Nonconventional (TMDNC) and TMDI tuned to higher modes or with a multi-level topology structure, a generalized MDOF model (see Fig. 9a) can be used for more reliable vibration control performance evaluation and parameterized design. Fig. 9b lists the performance influencing factors of TMDI.

In order to combine the optimization of TMDI tuning with the design of the main structure to improve performance under dynamic loads, the influence of the elastic and mass characteristics of the main structure on the performance of TMDI motion control is investigated, an innovative parameter study can be conducted. This relates to various tapered beam-like cantilever main structures which produce continuously varying bending stiffnesses and mass distributions [13,62–65].

By combining an inertial damper with Tuned Mass-Damper, an inertial-based TMD was obtained, and the closed-form displacement equations and power equations of generic tuned mass-damper-inerter (GTMDI) were derived [52]. The working mechanism of GTMDI was revealed and elucidated, and it was confirmed that GTMDI has dual benefits, namely reduced input energy and enhanced dissipation power effect. Fig. 10 is a schematic diagram of the coupling between the mechanical and motor network systems in the TMDI system.

Studies have found that adding electromagnetic motors to TMDIs with different damping characteristics can increase the available energy of wind-induced high-rise buildings [4]. In TMDI topologies where inertial units cover more floors, the acceleration of the floors can be simultaneously reduced while increasing the available energy. Theoretical studies have been conducted on the performance of non-linear shape memory alloy (SMA) springs in inertial devices for controlling wind-induced vibrations in high-rise buildings [6]. The potential for energy harvesting using this temporary passive control device was also investigated.

From the perspective of frequency domain analysis, compared to traditional TMDs, classical TMDs only tune the mechanical damper, while ERS-TMDI can improve vibration control and energy harvesting performance by tuning the resonance of mechanical dampers, inertial units, and electromagnetic resonators [5]. Therefore, ERS-TMDI can efficiently harvest energy and improve energy harvesting efficiency (see Fig. 11). Fig. 11a shows a typical generated fluctuating wind time history with a duration of 100 s and a time interval of 1 s. In Fig. 11b, the wind time history produced is compared to the Davenport spectrum by inverse fast fourier transform (IFFT) for power spectral density. It is clear from Fig. 11 that both classical TMD and ERS-TMDI have displacement damping effect at the same time and their relative reduction rates are 9.71% and 11.49%. In this case, the peak displacement of the primary structure of ERS-TMDI is reduced by about 9.78%, while the relative reduction rate in terms of acceleration is 6.95% [5].

Luo et al. [5] introduced the standard performance of ERS-TMDI by comparing it to classical TMD, ERS-TMDs, and Series Dual Mass TMDs (SDM-TMDs), in order to achieve minimum structural damage and energy harvesting under stochastic wind excitation. As shown in Fig. 12b, Petrini et al. [50] studied the potential of coupling TMDs with inertial devices in different TMDI topologies to dissipate the vibration due to crosswind vortex forces in high-rise buildings while generating electric energy. When a renewable electromagnetic motor (EM) is connected to the TMDI, the damping characteristics of TMDI will be allowed to change, and part of the kinetic energy dissipation will be converted into electrical energy. The results (see Fig. 12a) show that the energy available for collection can be increased by increasing the damping of the TMDI beyond the optimum value for vibration suppression, or by sacrificing greater bottom plate acceleration at the expense of reducing the inertial characteristics.

Since the optimal stiffness and damping characteristics do not significantly change with the reference wind intensity used in the design and evaluation of high-rise buildings for occupant comfort and accessibility, the potential for TMDI to simultaneously perform energy harvesting and vibration suppression has been quantified by coupling TMDI with regenerative electromagnetic motors, allowing for changes in optimal TMDI damping and inertia. Therefore, in wind-induced high-rise buildings, with regard to the power generation control of sensors and actuators for health monitoring and climate control in modern structures, this can be increased by considering the occupancy situation or relaxing the availability limit state thresholds for floor acceleration to meet the comfort of the occupants.

The considered TMDI configuration is suitable for controlling the dominant vibration mode of a vertical cantilever structure [66]. Based on this, an analytical model of a cantilevered flexural structure with an electromagnetic pulse as an ideal damper for TMDI is established. The main structure is considered as a generalized single-degree-of-freedom structure with mass and stiffness characteristics depending on the first vibrational mode. The analysis results show that under stationary white noise excitation, the further distance between the inertia mass and the top of the main structure, the better.

A large body of research has shown that ERS-TMDI has gained broad attention in the field of vibration mitigation and energy harvesting (see Fig. 13). Although this design is well established in linear structures, it has not been widely applied in nonlinear structures [44]. The significance of this work lies in the use of only a single device to achieve vibration suppression and energy harvesting of non-linear vibrating structures. The experimental results show that the proposed NERS-TMDI structure performs better than the existing non-linear tuned mass dampers (NTMD) and ERS-TMDI in terms of vibration control (see Fig. 13). The results also show that the first configuration of the NERS-TMDI always outperforms the second configuration of the NERS-TMDI in terms of vibration control and energy harvesting. However, the stability and bifurcation analysis of the NERS-TMDI remains to be investigated and future work will involve the development of suitable optimization schemes to improve the existing design options for the NERS-TMDI [48,49].

The electromagnetic tuned mass-inerter damper (EM-TMID), proposed by Luo et al. [67], achieves the dual functions of vibration suppression and energy harvesting. A new feature of this damper is that it replaces the viscous damping of the traditional TMD with an inertial electromagnetic transducer. Based on the Alembert principle, a dynamic model of the EM-TMID and a single-degree-of-freedom structure under earthquake excitation was established. Using the H2 norm method with the minimum root-mean-square damage value of the primary structure as the objective, the EM-TMID was optimized by considering both frequency domain and time domain numerical simulations, and the dual functions of vibration suppression and energy harvesting were analyzed. The results show that EM-TMID outperforms classical TMD, EM-TMD and TMDI in reducing the peak and area of the frequency response of the primary structure displacement in the frequency domain; while EM-TMID outperforms classical TMD in reducing the peak and root mean square values of the displacement and acceleration in the time domain. The mechanical model of the EM-TMID-single-degree-of-freedom structure coupling system can be established based on the D’Alembert principle as follows:

(4){msx¨s+csx˙s+ksxs−kT(xT−xs)+kfI=−msx¨g(mT+b)x¨T+kT(xT−xs)−kfI=−mTx¨gkv(x˙T−x˙s)+RI+LI˙=0

where kv, kf are the electromagnetic sensor constants, ideally kv = kf; R is the resistance, R = R_i + R_e; kT is the stiffness of the TMD; mT is the mass of the TMD; b is the inertia mass; xT is the displacement of the TMD; xs is the displacement of the main structure; and L is the electric current.

The traditional TMD is connected to the ground via an inertial device. The inertial electromagnetic transducer and its related circuitry replace the viscous damping of the classical TMD. The transducer is assumed to be an ideal transducer that is parallel to a resistor, an inductor, and a capacitor (RLC) circuit made up of resistance, inductance, and capacitance. Two types of RLC circuit structures were studied: one is in series and the other is in parallel. The equations of motion for the system are given as follows [3]:

msx¨s+csx˙+ksxs−kT(xT−xs)+kfI=−msx¨g

(5)(mT+b)x¨T+kT(xT−xs)−kfI=−(mT+b)x¨g

kv(x˙T−x˙s)+RI+LI˙+1c∫Idt=0

where kv is the transducer voltage constant. kf is the force constant of the transducer.

Fig. 14a is the representations of structures. The objective of ERS-TMDI is to simultaneously suppress unnecessary vibration and collect energy from vibrating structures (see Fig. 14b). The Matlab optimization toolbox was used to numerically verify the optimal expression obtained (see Fig. 14c). The results showed good consistency between the two. These optimal parameters, namely the electromagnetic-mechanical coupling coefficient, were found to be dependent on the mass ratio and inertia ratio. Through parameterized studies, the effects of the mass ratio and inertia ratio on the optimized design parameters were examined. The results showed that an increase in the mass ratio and inertia ratio can improve the electromagnetic-mechanical coupling coefficient and damping ratio, and it was also found that the parallel RLC structure (see Fig. 14b) is superior to the series RLC structure in energy harvesting and has similar performance in vibration suppression [3,67].

In conclusion, it can be seen that all studies on TMDI in the context of energy harvesting clearly demonstrate that the proposed inertial device is feasible and useful in controlling wind-induced vibration of buildings and converting dissipating energy into electrical energy.

Bian et al. [68] studied design, optimization, and evaluation techniques for TMDI used on circular cross-sectional structural components. Zero-mean white noise random processes were used as external excitation loads. Two design objectives were to minimize the variance of the primary structure’s displacement and to maximize the energy dissipation index (EDI) as a method of selecting the optimal design parameters. Lara-Valencia et al. [69] discussed the optimization design of TMDI using exhaustive search and demonstrated that linear combination optimization design schemes are valuable and easy-to-implement tools. It was suggested that as research and practical applications of TMDI devices continue to develop, nonlinear analysis should be the focus of future developments in this field. Masnata et al. [70] proposed a new type of TMDI, which incorporated a recently developed Non-Traditional Tuned Mass Damper (referred to as new TMD) in combination with a base-isolation (BI) system to control the displacement of a base-isolated structure under earthquake action (see Fig. 15). The new TMD consisted of a secondary mass system connected to the BI system’s baseplate via a spring and connected to the ground via a damper, as well as an inertial device parallel to the damper, as shown in the figure below. Ye et al. [71] used simplified 3DOF and MDOF structural models to evaluate the effectiveness of a double-isolation system with optimal supplementary disturbance sources by using artificial and actual seismic records for time-lapse analysis. The results show that the maximum absolute floor acceleration of the dual isolation system (DIS)-inerter-MDOF-1 model is significantly lower than that of the base-isolated structure of multi-degree-of-freedom (BIS-MDOF)-MDOF model, but slightly higher than that of the original double-isolated system.

This device is connected to the baseplate of the isolated structure via a spring and a damper in the traditional form of TMD. Some studies have shown that it can effectively reduce the displacement of the base-isolation sub-system and the primary structure [24,25]. This novel design includes a TMDI with the damping element directly connected to the ground. The results show that under the same mass ratio, compared to the classic TMDI, the new TMDI can better reduce the displacement of its substrate isolation subsystem even further [26–28].

Pietrosanti et al. [65] presented an optimization technique for designing TMDI for MDOF structures. By analyzing the influence of the mass ratio and comparing its performance with that of TMD with the same mass, it was found that using an inerter is particularly suitable when using a small amount of auxiliary mass. The optimization design of TMDI was investigated by di Matteo, who proposed a numerical method to directly optimize the TMDI under white noise excitation. This was done to control the acceleration response of a base-isolated structure. A. Giaralis established a design framework for earthquake-excited TMDI-assembly-layer buildings, considering mass amplification effects, using inertia as a design parameter, and considering the stiffness and damping characteristics of TMDI. By ensuring that the contribution of all structural modes to the design objective function evaluation is explicitly taken into account, optimal design is achieved across the entire frequency range [70,72,73]. It was found that compared with the same mass TMD, when the TMD mass and TMDI topology structure are smaller (where the inerter connects the TMD mass to the following two layers), including the inerter achieves enhanced performance improvement, and TMDI has quite robust properties, with limited influence of excitation characteristics on actual optimization design [74–76].

Djerouni et al. [77] developed a mathematical model for TMDI in building seismic mitigation using MATLAB Simulink, which demonstrated the robustness, performance, and effectiveness of TMDI. de Domenico et al. [78] found the optimal parameters for TMDI based on probability framework and investigated different optimization algorithms (see Fig. 15d). The effectiveness of the optimal TMDI parameters was evaluated through time history analysis of multi-story isolated structure under multiple earthquake excitations. Three distinct methods for optimizing TMDI were developed using white noise as the fundamental excitation. The efficiency of designing and assessing TMDI systems based on earthquake excitation was also explored [1]. The results showed that TMDI could effectively reduce the dynamic response of the system, and the use of inertial systems could achieve lightweight of the system, with better performance and robustness than classical systems. Two kinds of nonlinear constitutive behaviors were considered, and a dynamic layout scheme was proposed in which TMDI was installed below the isolation layer of the foundation isolation structure to improve the seismic capacity and reduce the displacement demand of the isolation structure [79]. Patsialis et al. [12] established a reduced order model (ROM) on the basis of the original finite element model. However, Kaveh et al. [41] computed the optimal parameters for the free vibration of TMDI with single and double inertias arranged at various positions in the optimization design of both SDOF and MDOF models (refer to Fig. 16). These parameters include the natural frequency and damping ratio. The results showed that the tuned dampers with rational configurations had better performance than the classical TMDs with the same weight (mass).

Firstly, a case study was conducted to apply the inertia dampers to adjacent high-rise buildings. Secondly, a multi-objective evolutionary algorithm, i.e., non-dominated sorting genetic algorithm, was used to solve the quality-constrained optimization problem of multi-tuned mass damper inerter (MTMDI) parameter [9,80]. Finally, a comparison was conducted among MTMD, single TMDI, and MTMDI systems that consist of TMDI and TID while considering practical design considerations that are related to using realistic and feasible constraints for maximum mass ratio. de Domenico proposed a TMDI-enhanced foundation isolation system mounted on an isolation floor [78,25]. The effectiveness of the optimal TMDI parameters was evaluated through time history analysis of multi-story isolated structure under multiple earthquake excitations. By applying D’Alembert’s principle, the motion control equation of a three-dimensional system can be easily expressed in matrix form [78]:

(6)Mu¨(t)+Cu(t)+Ku(t)=−τu¨g(t)

where g(t) is the horizontal ground acceleration to which the combined system is subjected.

Pietrosanti et al. [81] presented an optimization design method for TMDI based on multi-degree-of-freedom structures. The study examined MDOF structures with ground-connected or unconnected TMDIs and proposed a method for a generalized two-degree-of-freedom model suitable for optimizing control system design. Under the basic acceleration with only the horizontal translational component of motion considered at each level, the motion equations for such a structure can be expressed as follows [81]:

Mu¨+Lu˙+Ku=−Mτu¨g+BkfB+BjfT

(7)mTu¨T=−mTu¨g−fB−fT

fB=b(u¨T−u¨k)fT=cT(u˙T−u˙j)+kT(uT−uj)

where M, L, and K are the mass, damping, and stiffness matrices, respectively, u is the horizontal displacement vector with respect to the ground, uT is the relative displacement of the mass mT with respect to the ground, and uj, uk are the relative displacements of the j and K degrees of freedom with respect to the ground, respectively. τ is the unit vector, fB is the interfering force acting on k-dof, and fT is the viscoelastic force acting on j-dof, and finally Bk and Bj are the assigned vectors.

A comparison between traditional passive TMD and TMDI was made, and the building was modeled mathematically with MATLAB Simulink [1]. The simulation results demonstrated the effectiveness, robustness, and performance of the method. In addition, a new structure called Tuned Mass-Damper Inertial Resonator (TMDIR) was proposed to suppress ground motion, such as base or excitation force. Liu et al. [82] investigated the vibration control and optimization design problems of the Tuned Mass-Damper Inertial Transmission Line (TMDITL) and they developed the differential motion equation for TMDI transmission line under harmonic excitation and utilized frequency domain analysis to obtain a closed-form solution for the displacement response spectrum of the transmission line. The design parameters of TMDI were optimized using fixed point theory and the researchers discussed the vibration control performance of TMDI based on this optimization.

A dynamic layout scheme was proposed in which TMDI was installed below the isolation layer of the foundation isolation structure to improve the seismic capacity and reduce the displacement demand of the isolation structure [79,83]. Matteo et al. [74,84] studied the response of the foundation isolation structure controlled by TMDI under stochastic horizontal base acceleration. Firstly, a direct numerical method was proposed to optimize the device under white noise excitation, and a simplified analytical solution for TMDI parameter optimization design of foundation isolation structure was further proposed. Under this method optimization, the response of the foundation isolation structure under strong earthquake excitation can be effectively reduced. Bian et al. [85] proposed a hybrid vibration control method BI-TMDI for the vibration suppression of substation systems. A theoretical optimization model of the switches mounted on frame (SMF)-BI-TMDI system based on a three-degree-of-freedom analogy was developed, which can be used to explicitly consider the interactions between the switch, base isolation frame and TMDI. The results of the time course analysis of the SMF-BI-TMDI system under the action of a strong seismic ground motion ensemble validate the effectiveness of the BI-TMDI vibration control method as a promising vibration control method that can be used to improve the seismic response of the substation and its components.

The article introduced a calculation method for structural damage under stochastic ground acceleration excitation and compared it with the ERS-TMDs, TMDI, and TMD, as shown in Fig. 17 of the article referenced by citation [86]. The article mentioned that closed-form solutions were obtained, which included the optimal mechanical tuning ratio, optimal electrical damping ratio, optimal electrical tuning ratio, and optimal electro-magnetic-mechanical coupling coefficient. Fig. 17d in the article illustrated the optimal performance indicators for earthquake control.

Petrini et al. [4,87] proposed a new scheme for the optimization design problem of TMDI to determine the damping and stiffness characteristics of TMDI. This approach can minimize the key floor acceleration for building occupants in cross-wind direction using the previously designed TMDI inertia characteristics and topology structure (see Fig. 18a). Through extensive inertia characteristics and three different topology structures, optimized TMDIs were obtained (see Fig. 18b).

Three distinct optimization techniques were introduced for TMDI, utilizing a zero-mean white noise random process as the primary stimulus. A MTMDI system (consisting of multiple TMDIs) was used as an unconventional seismic protection strategy to connect two adjacent high-rise buildings [1]. Using NSGA-II for displacement-, inter-story drift-, and acceleration-oriented optimization, four optimal parameters for each TMDI in the MTMDI system, namely, mass ratio, inertia ratio, frequency ratio, and damping ratio were obtained [88]. Two inertia-based isolation systems (see Fig. 19), namely, the inertial damper (ID)-based isolation system, in which an isolation system based on a parallel connection of an ID and a viscous damper, as well as a series connection of an inertial damper, a spring damper, were put forward. The former is referred to as a TID-based isolation system [89]. Based on the optimal damping ratio and inertia ratio, both ID-based and TID-based isolation systems can reduce relative displacement, absolute acceleration, and base displacement. The results showed that ID-based and TID-based isolations have great prospects in the application of base isolation. A seismic hazard model was developed by Patsialis et al. [12] using a representative set of acceleration time course models to investigate the implementation of TMD and TID as limit cases for TMDI. The results show that both TID and TMDI achieve almost identical performance throughout the front end, for both linear and non-linear structures. When the inerter vibration absorber (IVA) is placed at a lower floor level, improved vibration control efficiency is achieved for both linear and non-linear structures, and better vibration suppression is achieved by allowing the IVA to span two floors. A novel design formula for a TMDI system was proposed in order to address the discomfort experienced by residents in tall and slim high-rise buildings that are susceptible to vortex shedding effects. Furthermore, the potential for converting wind-induced energy into usable electricity for these buildings was explored using this optimal TMDI design [19,90,91].

The majority of the research on TMDI systems has centered around optimizing tuning methods and connection arrangements to enhance the motion control efficacy of specific primary structures. Domenico and colleagues also investigated the impact that the elastic and mass properties of the primary structure have on the motion control performance of TMDI systems (see Fig. 20). The innovative parameter study conducted by Domenico and his team involved a vast array of conical beam-like cantilever primary structures, enabling the exploration of the combination of optimal TMDI tuning with primary structure design to enhance dynamic load performance. This represents a significant step forward in the field of TMDI systems research.

The advantages and disadvantages of the existing optimization methods are summarized and the following Table 2 shows the optimization methods for TMDI.

According to Zhu’s analysis, the transfer functions of displacement and acceleration responses suggest that TMDI is more efficient than TMD in reducing wind-induced responses of coupled high-rise buildings. This is true even when the mass of TMDI is one-third smaller than that of TMD. Application of TMDI to wind vibration control in coupled high-rise buildings. This would be a good solution to the wind vibration control problem in coupled high-rise buildings. In the following discussion and research, we will focus on the relevant hot issues of TMDI in the field of high-rise buildings.

Research on vibration control of bridges under the condition of non-ideal vortex-induced vibration based on inertial conditions is quite limited in actual large-span bridge engineering. Since Jin et al. first proposed using Inerter-based dynamic vibration absorbers (IDVAs) to reduce bridge vibration under harmonic excitation in 2016, Dai et al. [98] mentioned that under the same mass ratio, IDVA is more efficient than traditional TMD. Xu et al. later proposed using TMDI to suppress vortex-induced vibration of bridges in 2019 (see Fig. 22b). Since the inertial body of TMDI has an inertia force, which is equal to adding a virtual mass on the basis of the vibration mass block of the original TMD, TMDI has an excellent effect on the seismic control of bridges.

In recent years, the research conducted by Dai et al. in 2019 focused on the effect of inertial span on TMDI control efficiency, while Xu et al. compared different IDVA layouts and their effects on vortex-induced vibration (VIV) control in large-span bridges in 2020. Dai et al. utilized the Maxwell element to replace the damping element in TMDI in 2021 for VIV control in bridges. However, these are only theoretical studies based on TMDI. In this article, we will summarize the implementation and optimization of vibration control using TMDI in large-span bridges [99]. The schematic diagram of the TMDI system in bridge construction is shown in Fig. 22 below [95].