# Wind Induced Vibration Control and Energy Harvesting of Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter for Building Structures

Authors: Yifan Luo; Hongxin Sun; Xiuyong Wang; Lei Zuo; Ning Chen
Journal: Shock and Vibration
 (2017)

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## Abstract

This paper proposes a novel inerter-based dynamic vibration absorber, namely, electromagnetic resonant shunt tuned mass-damper-inerter (ERS-TMDI). To obtain the performances of the ERS-TMDI, the combined ERS-TMDI and a single degree of freedom system are introduced.H2 criteria performances of the ERS-TMDI are introduced in comparison with the classical tuned mass-damper (TMD), the electromagnetic resonant shunt series TMDs (ERS-TMDs), and series-type double-mass TMDs with the aim to minimize structure damage and simultaneously harvest energy under random wind excitation. The closed form solutions, including the mechanical tuning ratio, the electrical damping ratio, the electrical tuning ratio, and the electromagnetic mechanical coupling coefficient, are obtained. It is shown that the ERS-TMDI is superior to the classical TMD, ERS-TMDs, and series-type double-mass TMDs systems for protection from structure damage. Meanwhile, in the time domain, a case study of Taipei 101 tower is presented to demonstrate the dual functions of vibration suppression and energy harvesting based on the simulation fluctuating wind series, which is generated by the inverse fast Fourier transform method. The effectiveness and robustness of ERS-TMDI in the frequency and time domain are illustrated.


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## Body

## 1. Introduction

It is a well-known fact that protecting structures against dynamic loads such as winds and earthquakes has been one of the chief purposes in structural design. Many research has been conducted to search for complementary energy dissipation devices to suppress the harmful oscillation. Among these methods, the classical tuned mass-damper (TMD) [1], as dynamic vibration absorber, has been proved to be one of the most effective and popular supplemental energy dissipation devices [2–5], with many pragmatic installations [6–8].Later on, various methods to optimize the parameters of TMD have been developed. For the TMD with undamped primary systems, there are some common methods, such as fixed-point method,H2 norm, and H∞ optimal method. For the TMD with damped primary systems, substantive design methods and tuning criteria have been raised [9], and the TMD applied in nonlinear and distributed primary systems has been studied [10, 11].Meanwhile, energy harvesting from large-amplitude low-frequency oscillating primary structures has emerged as a promising research area [12, 13], especially from TMD [14–19]. It depends on replacing the energy-dissipating element of the TMD or supplementing with electromagnetic harvester, for relatively large-scale applications [14–18], or piezoelectric materials, even for relatively small-scale applications [17, 19]. Under this background, a concept of tuned mass-damper/harvester (TMD/H) is presented in [20], in which the basic conversion consists of a linear voice coil motor connected to a resistance emulator consisting of rectification and variable impedance unit. Along a similar path, an electromagnetic transducer connected with an energy harvesting-enabled circuit is used to transform vibration energy into electric power and also to supply controlled force vibrations mitigation for TMD-equipped multilayer buildings structures in [11]. In addition, Ali and Adhikari proposed the idea of an energy harvester dynamic vibration absorber (EH-DVA) [19], in which electric energy is generated from strains developed in layers of piezoelectric material with the electric circuit mounted onto the attached vibrating mass of a TMD. They also provided an analytical derivation of the EH-DVA under harmonic and random excitation [19, 21].Lately, an inerter, a two-terminal mechanical device, is proposed by Smith et al. [22, 23]. The inerter is a device that provides a force proportional to the relative acceleration between its two terminals. Moreover, the inerter has been studied for improvement of car suspensions [24–26] and railway vehicles [27, 28]. Other applications of inerters can also be found in civil engineering for improving the performance of vibration control and displacement mitigation in base isolation systems. It was proposed the idea of using an inerter-like ball-screw mechanism to be employed in a viscous mass-damper or tuned viscous mass-damper system in [29]. A type of electromagnetic inertial mass-damper using a ball-screw mechanism and a motor was presented to control the vibration of structures subjected to earthquakes in [30]. Moreover, an inertia-based passive control based on a gyro-mass-damper using gear assemblies was also studied as a type of supplemental damper [31] and as a base isolation system [32]. Lazar et al. [33, 34] proposed an inerter-based system with a configuration similar to that of a TMD, which is termed as tuned inerter damper (TID) and developed an analytical tuning rule of a TID based on the Den Hartog’s method. Marian and Giaralis [35] also proposed the concept of tuned mass-damper-inerter (TMDI) to mitigate the oscillatory motion of support systems which are stochastically excited. They also developed the optimal TMDI parameters for undamped primary structures under white noise excitation in closed-form as functions of the TMD mass and the inertance. Further, they proposed the energy harvesting-enabled tuned mass-damper-inerter (EH-TMDI) configuration for simultaneous vibration suppression and energy harvesting [36].Inspired by the idea of EH-TMDI [35, 36], this paper proposes a novel inerter-based dynamic vibration absorber, called electromagnetic resonant shunt tuned mass-damper-inerter (ERS-TMDI), in which we replace the dissipated element of the TMDI with the electromagnetic transducer shunted with a resonant R-L-C circuit for simultaneous vibration control and energy harvesting. In addition, this paper derives the closed-form solution of optimal design parameters to the ERS-TMDI for building structures under random wind and presented numerical analyses. H2 norm optimal method is employed for the undamped single degree of freedom (SDOF) structure, with the objective to minimize the mean squared value of the displacement of the primary structure. The advantages of the ERS-TMDI will be illustrated through the comparison with the classical TMD, series-type double-mass TMDs [37], and ERS-TMDs [38].The remainder of this paper is organized as follows. In Section2, the governing equations of the SDOF system with ERS-TMDI are derived. Section 3 derives optimal H2 closed-form solutions to the ERS-TMDI systems under the wind force excitation. In Section 4, the numerical analysis of structures is presented in comparison with the classical TMD and ERS-TMDs. Conclusions are drawn in Section 5.


## 2. The Single Degree of Freedom System Model of the Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter

As is shown in Figure1(a), for easily understanding the reduction effects of the damper, a multistory building is simplified as a general multidegree of freedom (MDOF) system. Further, Figure 1(a) also shows the configuration of ERS-TMDI for the MDOF, where the ERS-TMDI is obtained by replacing the original energy dissipative damping cT in classical TMD with an electromagnetic transducer shunted with a resistance R, capacitance C, and inductance L [16, 39], and an inertanceb, a two-terminal mechanical device, is assigned between the absorber and next story of structure, for simultaneous vibration control and energy harvesting under the force excitation of the primary structure.Figure 1
Schematic of the coupled structure and tuning damper system: (a) a general MDOF and (b) the SDOF structure-damper system.When multistory buildings are subjected to wind vibration, the first-order modal played a very important role in the analysis of vibration. Therefore, the primary structure is characterized as single degree of freedom (SDOF) structure with a linear spring of stiffnessks, a mass ms, and a damping coefficient cs, as shown in Figure 1(b).The motion equations of the combined SDOF and ERS-TMDI system, shown in Figure1(b), can be given by(1)msx¨s+csx˙s+ksxs-kTxT-xs+kfI=F,mT+bx¨T+kTxT-xs-kfI=0,kvxT˙-xs˙+RI+LI˙+1C∫Idt=0;when the primary structure is undamped, the above can be simplified in the Laplace domain:(2)msxss2+ksxs-kTxT-xs+kfI=F,mT+bxTs2+kTxT-xs-kfI=0,kvxT-xss+RI+LIs+ICs=0;normalizing the frequency and setting s=jω, we can have(3)xsjα2+xs-μfT2xT-xs+kfksI=Fks,1+δxTjα2+fT2xT-xs-kfμksI=0,kvxT-xsjα+2ζefeLI+LIjα+ILfe21jα=0;rewriting relative displacement xr=xT-xs,(4)xsjα2+1-μfT2xr+kfksI=Fks,xr1+δjα2+fT2+xs1+δjα2-kfμksI=0,kvxrjα+LI2ζefe+jα+fe21jα=0,where the nature frequency of the primary structure ωs=ks/ms, the nature frequency of the absorber ωT=kT/mT, the resonant frequency of the circuit ωe=1/LC, the mass ratio of the absorber to the primary structure μ=mT/ms, the mass ratio of inertance to the primary structure δ=b/mT, the mechanical tuning ratio fT=ωT/ωs, the electrical tuning ratio fe=ωe/ωs, the normalized frequency α=ω/ωs, and the electrical damping ratio of the circuit ζe=R/(2Lωe), where the total resistance R=Ri+Re and Re is the equivalent external load; the electromagnetic mechanical coupling coefficient μk=kvkf/(kTL), where kv and kf are the voltage constant and the force constant of the electromagnetic transducer.


## 3.H2 Optimization for the ERS-TMDI

### 3.1. Vibration Mitigation of the ERS-TMDI

If the system suffered from the wind force rather than the sinusoidal excitation,H2 optimization is more desirable for evaluating the system performance for it is the root mean square (RMS) value of the performance under unit Gaussian white noise input. This is a good approximation when the frequency of the wind force is broad when compared with the natural frequency of the building. For mitigating the vibration of the primary structure when subjected to the excitation force F, H2 norm is minimized from F/ks to the deformation of the primary structure and xs is defined as(5)PIv=Exs22πωsS0=xs22πωsS0,where the symbols E[·] and · refer to the ensemble and temporal averages, respectively, and S0 is the uniform power spectrum function. The RMS value of the deformation of the SDOF structure mass xs can be obtained as(6)xs2=ωsS0∫-∞∞Xn2dα,where Xn in (6) is the norm of the frequency response function from F/ks to the deformation of the primary structure xs and j=-1 is the unit imaginary number. Substituting (6) into (5), the performance index PIv in (5) can be expressed as(7)PIv=12π∫-∞∞Xn2dα.In addition, the normalized frequency response function Xn from F/ks to xs can be written in the dimensionless form by using the aforementioned dimensionless parameters, which is(8)Xn=xsF/ks=fe2fT2+2fefT2ζejα+fe2ψ+fT21+μkjα2+2feψζejα3+ψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6,A0=fe2fT2,A1=2fT2feζe,A2=fe2ψ+fT21+μk+fe2fT21+μψ,A3=2feζeψ+fT21+μψ,A4=fT21+μk1+μψ+1+fe2ψ,A5=2feζeψ,A6=ψ,where ψ=1+δ.The integral in (7) can be solved using the general formula in [40]. Hence, the performance index in (7) can be obtained as a function of the four design parameters fT, μk, fe, ζe and the given parameters μ, ψ.(9)PIv=14fefT2μkζeμψ2ψ2+fe4ψ2+fe2ψ2-2+4ζe2+fT2-21+μkψ-2fe4ψ1+μψ+fe2ψ2+21+μk+μψ+μ1+μkψ-8ζe2-4μψζe2+fT41+μk2+μ1+μk2ψ+fe41+3μψ+3μ2ψ2+μ3ψ3+fe2-21+μk-4μ1+μkψ-2μ21+μkψ2+4ζe2+8μψζe2+4μ2ψ2ζe2.To minimize the performance indexPIv regarding the vibration mitigation performance, the derivatives of PIv with respect to all the design parameters should be equal to zero. Thus, we have(10)∂PIv∂fT=0,∂PIv∂ζe=0,∂PIv∂fe=0,∂PIv∂μk=0.Therefore, the following simultaneous gradients’ equations can be obtained from (10):(11a)fT41+μk21+μψ+fe4fT41+μψ3-2fe2fT41+μψ21+μk-2ζe2-ψ21+fe4+fe2-2+4ζe2=0,(11b)-2fT21+μkψ+fT41+μk2-2fe4fT2ψ1+μψ+fe4fT41+μψ3-2fe2fT41+μψ21+μk+2ζe2+fe2fT2ψ2+μψ2+μk+4ζe2+ψ21+fe4-2fe21+2ζe2=0,(11c)2fT21+μkψ+-fT41+μk2-6fe4fT2ψ1+μψ+3fe4fT41+μψ3+fe2fT2ψ2+μψ2+μk-4ζe2-2fe2fT41+μψ21+μk-2ζe2+ψ2-1+3fe4+fe2-2+4ζe2=0,(11d)-2fT2ψ+fT41-μk2-2fe4fT2ψ1+μψ+fe4fT41+μψ3+2fe2fT41+μψ2-1+2ζe2-2fe2fT2ψ2+μψ-1+2ζe2+ψ21+fe4+fe2-2+4ζe2=0.Solving this set of equations is nontrivial as it involves multiple nonlinear, high-order variables, but in the following we will simply summarize the solving process and then directly present the final results. This will highlight the results and avoid prolixity. By combining (11d) with the other three equations in (11a), (11b), (11c), and (11d) we can eliminate ζe to obtain a new equation set in design variables fT, μk, fe. Then using similar manipulations, we can also eliminate fT and fe from the new equation set and obtain (11a), (11b), (11c), and (11d) in only variable μk. Similarly, H2 norms optimal can be obtained by (11a), (11b), (11c), and (11d) as(12)fTopt=ψr-μψ21+μψ,μkopt=4μψ4+6μψ+r16+19μψ,feopt=32+25μ2ψ2+μψ58+r216+35μψ+19μ2ψ2,ζeopt=1/16+19μψμψ70μ3ψ3+244+r+4μψ68+11r+μ2ψ2247+18r16+23μψ+8μ2ψ2,where r=16+32μψ+17μ2ψ2.AtH2 optimal tuning condition, the performance index PIvopt is (13)PIvopt=-8+23μψ/2+4μ2ψ2784μ4ψ4+1924+r+20μψ156+31r+μ3ψ33147+232r+μ2ψ24716+661r21+μψ16+19μψμψ-r4+6μψ+r32+25μ2ψ2+μψ58+r/16+35μψ+19μ2ψ2μψ70μ3ψ3+244+r+4μψ68+11r+μ2ψ2247+18r/16+19μψ.Because the mass ratioμ is a pretty small number, regularly less than 0.1 for building-TMD systems, the square root term r=16+32μψ+17μ2ψ2 can be about as 4(1+μψ) by segmentally ignoring the terms involving second powers in μψ. Hence, a compendious, approximate solution set can be obtained as (14)fTopt∗=ψ4+3μψ21+μψ,μkopt∗=4μψ8+10μψ16+19μψ,feopt∗=32+29μ2ψ2+62μψ216+35μψ+19μ2ψ2,ζeopt∗=1/16+19μψμψ142μ3ψ3+495μ2ψ2+544μψ+19216+23μψ+8μ2ψ2with the approximate optimal performance index PIvopt∗,(15)PIvopt∗=16+23μψ+8μ2ψ21536+6368μψ+9840μ2ψ2+6719μ3ψ3+1712μ4ψ441+μψ4+3μψ4+5μψ16+19μψ64+124μψ+58μ2ψ2/16+35μψ+19μ2ψ2μψ192+544μψ+495μ2ψ2+142μ3ψ3/16+19μψ.Therefore, it is easy to obtain the corresponding optimal absorber stiffnesskTopt, the inductance Lopt, the capacitance Copt, and the total resistance Ropt with the above parameters,(16)Lopt=kvkfμkoptfTopt2mTωs2,kTopt=fTopt2mTωs2,Ropt=2kvkfζeoptfeoptμkoptfTopt2mTωs,Copt=mTμkoptfTopt2kvkffeopt2.Moreover, there is the need to declare thatH2 tuning law for the classical TMD system is [41](17)fopt=11+μ2+μ2,ζopt=μ4+3μ81+μ2+μ.Moreover, the approximate H2 tuning law for the ERS-TMDs is [42](18)fTopt∗∗=4+3μ21+μ,μkopt∗∗=32μ+40μ216+19μ,feopt∗∗=32+62μ+29μ2216+35μ+19μ2,ζeopt∗∗=μ192+544μ+495256-96μ-27μ2.And the optimal parameters of the series-type double-mass TMDs [37] are(19)μBopt=2μ,νopt=1+2μ,νBopt=11+2μ,ζ2opt=0,ζ3opt=123μ1+2μ,where μBopt is the mass ratio of the further mass to the closer mass; νopt is the undamped natural frequency ratio of the closer mass to the primary system; νBopt is the undamped natural frequency ratio of the further mass to the closer mass; ζ2opt is the damping ratio of the closer mass; ζ3opt is the damping ratio of the closer mass.In addition, the normalized frequency response functionXr from F/ks to the relative displacement of the primary structure xr can also be expressed in the dimensionless form by using the aforementioned dimensionless parameters, which are(20)Xr=xrF/ks=-fe2ψjα2+2ψfeζejα3+ψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6.


### 3.2. Energy Harvesting of the Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter

To optimize the energy harvesting of the ERS-TMDI, it is our wish to maximize the average electrical power on the external loadRe. The instant power on the external load is(21)P=ReI2.The normalized frequency response function fromF/ks to I can be written in the dimensionless form by using the aforementioned dimensionless parameters, which are(22)In=IF/ks=kv/Lψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6;similarly, the normalized frequency response function from F/ks to P can also be written:(23)Pn=PF/ks=InRex¨g/ωs2.At the optimal H2 tuning condition, the performance index PIe is (24)PIe=kv24fefT2μkζeμ1+δL2.


## 3.1. Vibration Mitigation of the ERS-TMDI

If the system suffered from the wind force rather than the sinusoidal excitation,H2 optimization is more desirable for evaluating the system performance for it is the root mean square (RMS) value of the performance under unit Gaussian white noise input. This is a good approximation when the frequency of the wind force is broad when compared with the natural frequency of the building. For mitigating the vibration of the primary structure when subjected to the excitation force F, H2 norm is minimized from F/ks to the deformation of the primary structure and xs is defined as(5)PIv=Exs22πωsS0=xs22πωsS0,where the symbols E[·] and · refer to the ensemble and temporal averages, respectively, and S0 is the uniform power spectrum function. The RMS value of the deformation of the SDOF structure mass xs can be obtained as(6)xs2=ωsS0∫-∞∞Xn2dα,where Xn in (6) is the norm of the frequency response function from F/ks to the deformation of the primary structure xs and j=-1 is the unit imaginary number. Substituting (6) into (5), the performance index PIv in (5) can be expressed as(7)PIv=12π∫-∞∞Xn2dα.In addition, the normalized frequency response function Xn from F/ks to xs can be written in the dimensionless form by using the aforementioned dimensionless parameters, which is(8)Xn=xsF/ks=fe2fT2+2fefT2ζejα+fe2ψ+fT21+μkjα2+2feψζejα3+ψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6,A0=fe2fT2,A1=2fT2feζe,A2=fe2ψ+fT21+μk+fe2fT21+μψ,A3=2feζeψ+fT21+μψ,A4=fT21+μk1+μψ+1+fe2ψ,A5=2feζeψ,A6=ψ,where ψ=1+δ.The integral in (7) can be solved using the general formula in [40]. Hence, the performance index in (7) can be obtained as a function of the four design parameters fT, μk, fe, ζe and the given parameters μ, ψ.(9)PIv=14fefT2μkζeμψ2ψ2+fe4ψ2+fe2ψ2-2+4ζe2+fT2-21+μkψ-2fe4ψ1+μψ+fe2ψ2+21+μk+μψ+μ1+μkψ-8ζe2-4μψζe2+fT41+μk2+μ1+μk2ψ+fe41+3μψ+3μ2ψ2+μ3ψ3+fe2-21+μk-4μ1+μkψ-2μ21+μkψ2+4ζe2+8μψζe2+4μ2ψ2ζe2.To minimize the performance indexPIv regarding the vibration mitigation performance, the derivatives of PIv with respect to all the design parameters should be equal to zero. Thus, we have(10)∂PIv∂fT=0,∂PIv∂ζe=0,∂PIv∂fe=0,∂PIv∂μk=0.Therefore, the following simultaneous gradients’ equations can be obtained from (10):(11a)fT41+μk21+μψ+fe4fT41+μψ3-2fe2fT41+μψ21+μk-2ζe2-ψ21+fe4+fe2-2+4ζe2=0,(11b)-2fT21+μkψ+fT41+μk2-2fe4fT2ψ1+μψ+fe4fT41+μψ3-2fe2fT41+μψ21+μk+2ζe2+fe2fT2ψ2+μψ2+μk+4ζe2+ψ21+fe4-2fe21+2ζe2=0,(11c)2fT21+μkψ+-fT41+μk2-6fe4fT2ψ1+μψ+3fe4fT41+μψ3+fe2fT2ψ2+μψ2+μk-4ζe2-2fe2fT41+μψ21+μk-2ζe2+ψ2-1+3fe4+fe2-2+4ζe2=0,(11d)-2fT2ψ+fT41-μk2-2fe4fT2ψ1+μψ+fe4fT41+μψ3+2fe2fT41+μψ2-1+2ζe2-2fe2fT2ψ2+μψ-1+2ζe2+ψ21+fe4+fe2-2+4ζe2=0.Solving this set of equations is nontrivial as it involves multiple nonlinear, high-order variables, but in the following we will simply summarize the solving process and then directly present the final results. This will highlight the results and avoid prolixity. By combining (11d) with the other three equations in (11a), (11b), (11c), and (11d) we can eliminate ζe to obtain a new equation set in design variables fT, μk, fe. Then using similar manipulations, we can also eliminate fT and fe from the new equation set and obtain (11a), (11b), (11c), and (11d) in only variable μk. Similarly, H2 norms optimal can be obtained by (11a), (11b), (11c), and (11d) as(12)fTopt=ψr-μψ21+μψ,μkopt=4μψ4+6μψ+r16+19μψ,feopt=32+25μ2ψ2+μψ58+r216+35μψ+19μ2ψ2,ζeopt=1/16+19μψμψ70μ3ψ3+244+r+4μψ68+11r+μ2ψ2247+18r16+23μψ+8μ2ψ2,where r=16+32μψ+17μ2ψ2.AtH2 optimal tuning condition, the performance index PIvopt is (13)PIvopt=-8+23μψ/2+4μ2ψ2784μ4ψ4+1924+r+20μψ156+31r+μ3ψ33147+232r+μ2ψ24716+661r21+μψ16+19μψμψ-r4+6μψ+r32+25μ2ψ2+μψ58+r/16+35μψ+19μ2ψ2μψ70μ3ψ3+244+r+4μψ68+11r+μ2ψ2247+18r/16+19μψ.Because the mass ratioμ is a pretty small number, regularly less than 0.1 for building-TMD systems, the square root term r=16+32μψ+17μ2ψ2 can be about as 4(1+μψ) by segmentally ignoring the terms involving second powers in μψ. Hence, a compendious, approximate solution set can be obtained as (14)fTopt∗=ψ4+3μψ21+μψ,μkopt∗=4μψ8+10μψ16+19μψ,feopt∗=32+29μ2ψ2+62μψ216+35μψ+19μ2ψ2,ζeopt∗=1/16+19μψμψ142μ3ψ3+495μ2ψ2+544μψ+19216+23μψ+8μ2ψ2with the approximate optimal performance index PIvopt∗,(15)PIvopt∗=16+23μψ+8μ2ψ21536+6368μψ+9840μ2ψ2+6719μ3ψ3+1712μ4ψ441+μψ4+3μψ4+5μψ16+19μψ64+124μψ+58μ2ψ2/16+35μψ+19μ2ψ2μψ192+544μψ+495μ2ψ2+142μ3ψ3/16+19μψ.Therefore, it is easy to obtain the corresponding optimal absorber stiffnesskTopt, the inductance Lopt, the capacitance Copt, and the total resistance Ropt with the above parameters,(16)Lopt=kvkfμkoptfTopt2mTωs2,kTopt=fTopt2mTωs2,Ropt=2kvkfζeoptfeoptμkoptfTopt2mTωs,Copt=mTμkoptfTopt2kvkffeopt2.Moreover, there is the need to declare thatH2 tuning law for the classical TMD system is [41](17)fopt=11+μ2+μ2,ζopt=μ4+3μ81+μ2+μ.Moreover, the approximate H2 tuning law for the ERS-TMDs is [42](18)fTopt∗∗=4+3μ21+μ,μkopt∗∗=32μ+40μ216+19μ,feopt∗∗=32+62μ+29μ2216+35μ+19μ2,ζeopt∗∗=μ192+544μ+495256-96μ-27μ2.And the optimal parameters of the series-type double-mass TMDs [37] are(19)μBopt=2μ,νopt=1+2μ,νBopt=11+2μ,ζ2opt=0,ζ3opt=123μ1+2μ,where μBopt is the mass ratio of the further mass to the closer mass; νopt is the undamped natural frequency ratio of the closer mass to the primary system; νBopt is the undamped natural frequency ratio of the further mass to the closer mass; ζ2opt is the damping ratio of the closer mass; ζ3opt is the damping ratio of the closer mass.In addition, the normalized frequency response functionXr from F/ks to the relative displacement of the primary structure xr can also be expressed in the dimensionless form by using the aforementioned dimensionless parameters, which are(20)Xr=xrF/ks=-fe2ψjα2+2ψfeζejα3+ψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6.


## 3.2. Energy Harvesting of the Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter

To optimize the energy harvesting of the ERS-TMDI, it is our wish to maximize the average electrical power on the external loadRe. The instant power on the external load is(21)P=ReI2.The normalized frequency response function fromF/ks to I can be written in the dimensionless form by using the aforementioned dimensionless parameters, which are(22)In=IF/ks=kv/Lψjα4A0+A1jα+A2jα2+A3jα3+A4jα4+A5jα5+A6jα6;similarly, the normalized frequency response function from F/ks to P can also be written:(23)Pn=PF/ks=InRex¨g/ωs2.At the optimal H2 tuning condition, the performance index PIe is (24)PIe=kv24fefT2μkζeμ1+δL2.


## 4. Numerical Analyses

### 4.1. Frequency-Domain Analyses of the ERS-TMDI

#### 4.1.1. Comparing ofH2 Tuning Laws of Exact and Approximate Solutions

Figure2 graphically depicts H2 tuning laws for the vibration mitigation when the primary structure is subjected to the force acceleration. It is obvious to see that the error between the exact solution and the approximate solution for the force excitation system is extremely small. Therefore, the approximate solution is a good alternative to avoid computational complexities in practice.Figure 2
Graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT; (b) optimal electromagnetic mechanical coupling coefficient μk; (c) optimal electrical tuning ratio fe; (d) optimal electrical damping ratio ζe of exact and approximate solutions under different mass ratios.
(a)
(b)
(c)
(d)


#### 4.1.2. Three-Dimensional Graphical Representations ofH2 Tuning Laws for Approximate Solutions

Figure3 graphically depicts H2 tuning laws of approximate solutions in a three-dimensional way for the vibration mitigation when the primary structure is subjected to the force excitations.Figure 3
Three-dimensional graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT, (b) optimal electromagnetic mechanical coupling coefficient μk, (c) optimal electrical tuning ratio fe, and (d) optimal electrical damping ratio ζe of approximate solutions.
(a)
(b)
(c)
(d)


#### 4.1.3. Comparing Performances of Different Systems

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of five types of SDOF systems are compared under the same mass ratio of 0.02. From Figure 4, it is clear that the ERS-TMDI is superior to mitigate the vibration of the primary structure nearly across the whole frequency spectrum for force excitation system, as compared to that of classical TMD, ERS-TMDs, and system without a TMD, and the effective frequency band is also further widened. At their own resonant frequencies, the peak value of the normalized displacement Xn in the ERS-TMDI system is reduced by around 35% compared to that of the classical TMD, 18% to that of the ERS-TMDs, and 17% to that of series-type double-mass TMDs, while in the normalized displacement Xr, the relative reduction ratio to classical TMD and ERS-TMDs is 36% and 33%.Figure 4
Schematic of the optimal frequency responses for types of SDOF system, where the mass ratioμ=0.02. (a) The frequency response of the deformation of the primary structure and (b) the relative deformation of different buildings.
(a)
(b)


#### 4.1.4. Comparing Performances of the ERS-TMDI under Different Inertance Ratios

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of the ERS-TMDI are compared under different inertance ratios. From Figure 5, H2 norm of the ERS-TMDI is better at the highest inertance ratio.Figure 5
Schematic of the optimal frequency responses under different inertance ratio: (a) the deformation of the primary structure and (b) the relative deformation of the ERS-TMDI.
(a)
(b)


### 4.2. Time-Domain Analyses of the ERS-TMDI

In this section, the Taipei 101 tower will be taken as a case study to be computed for simulated wind in order to demonstrate the dual effect of the ERS-TMDI. Results are presented in comparison with the classical TMD.


#### 4.2.1. Parameters of Simulation

In this section we take the Taipei 101 tower as a case study and illustrate the dual functions of ERS-TMDI. Results are presented in comparison with the classical TMD.Taipei 101 is one of the tallest buildings in the world (449.2 m to roof and 509.2 m to spire). A TMD of 660 tonnes is suspended on the top of the building from the 92nd to the 87th floor to suppress the wind induced vibration [43]. The TMD is 0.78% of the modal mass, the first natural frequency is 0.146 Hz, and inherent damping of the building is 1%. In the case study, the parameters of the classical TMD are designed using H2 optimization as shown Table 1.Table 1
The parameters of simulation.
Description
Symbol
Value
Primary structure mass
m
s
8.46× 107 [kg]
Mechanical TMD mass
m
T
6.60× 105 [kg]
Inertance mass
b
3.30× 105 [kg]
Stiffness of the primary structure
k
s
7.12× 107 [N/m]
Stiffness of the mechanical TMD
k
T
8.21× 105 [N/m]
Total inductance of electrical resonator
L
1.17 [H]
Total capacitance of electrical resonator
C
1.02 [F]
Internal resistance of linear motor
R
i
0.1 [Ω]
External resistance
R
e
0.1 [Ω]
The constant of the electromagnetic transducer
k
v, kf
150


#### 4.2.2. Wind Simulation

When the wind effect is analyzed, the Davenport wind spectrum expressed by (25a), (25b), and (25c) is taken as the target wind spectrum of wind velocity field [44]. The inverse fast Fourier transform method is used to generate the random fluctuating wind time series [45]. (25a)Sω=8πn2u∗2ω1+n24/3,(25b)n=600ωπV10,(25c)u∗=VzKln⁡z/z0,where Sω is the power spectrum of velocity fluctuation; ω is the frequency of the fluctuating wind; K is the terrain roughness factor of this area, which is 0.03; V10 is the mean wind speed, in m/s, at 10 m. It is taken as 39.93 m/s, which means that 50-year return period brings averaged over 10 minutes at a height of 10 m [46]. u∗ is the shear velocity of the flow, K=0.4. z is the height above the surface, z0 is the roughness length, and Vz is the mean wind speed of the height z.Figure6(a) is the typical fluctuating wind time history generated, whose duration is 1000 s and the time interval is 0.1 s. As shown in Figure 6(b), the generated wind time history is converted to power spectral density by IFFT, which is compared with the target Davenport spectrum. It is shown that the simulated spectrum is in agreement with Davenport spectrum well in most frequency bands.Figure 6
Simulated fluctuating wind time history: (a) wind speed and (b) power spectrum.
(a)
(b)With the information above, one can calculate fluctuating wind force by the following expression:(26)Ft=ρ·CD·A·V10·ut,where ρ=1.28 kg/m3 is the air density, CD=1.2 is the drag coefficient, the tributary area assuming is A=1600 m2, and ut is the fluctuating wind speed.


#### 4.2.3. The Performance of Vibration Mitigation

In the time-history analyses, the characteristics of the model are identical to those used in the frequency-domain analyses described in Section4.1.From Figure7, it is clear that the ERS-TMDI sufficiently reduces the displacement of the primary structure. In addition, the performance of the vibration mitigation of the ERS-TMDI is clearly superior to that of the classical TMD. The peak value of the displacement of the primary structure specially is reduced by around 9.78% compared to that of the classical TMD, while, in accelerations, the relative reduction ratio is 6.95%. On the other hand, in the root mean square value of the primary structure displacement, the relative reduction is 9.71% and 11.49%.Figure 7
Time-history responses of (a) displacements and (b) accelerations of the primary structure: comparison between classical TMD and ERS-TMDI.
(a)
(b)


#### 4.2.4. The Performance of Energy Harvesting

From Figure8 and Table 2, it is clear that the ERS-TMDI can also adequately harvest the power, whose peak power, average power, and total energy harvested are 1.34 × 103 w, 132.32 w, and 1.39 × 105 J, respectively.Table 2
The energy-related indicator.
Peak power/w
Average power/w
Energy harvested /J
1.34 × 103
132.32
1.39 × 105Figure 8
Time histories of the harvested power of ERS-TMDI.


## 4.1. Frequency-Domain Analyses of the ERS-TMDI

### 4.1.1. Comparing ofH2 Tuning Laws of Exact and Approximate Solutions

Figure2 graphically depicts H2 tuning laws for the vibration mitigation when the primary structure is subjected to the force acceleration. It is obvious to see that the error between the exact solution and the approximate solution for the force excitation system is extremely small. Therefore, the approximate solution is a good alternative to avoid computational complexities in practice.Figure 2
Graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT; (b) optimal electromagnetic mechanical coupling coefficient μk; (c) optimal electrical tuning ratio fe; (d) optimal electrical damping ratio ζe of exact and approximate solutions under different mass ratios.
(a)
(b)
(c)
(d)


### 4.1.2. Three-Dimensional Graphical Representations ofH2 Tuning Laws for Approximate Solutions

Figure3 graphically depicts H2 tuning laws of approximate solutions in a three-dimensional way for the vibration mitigation when the primary structure is subjected to the force excitations.Figure 3
Three-dimensional graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT, (b) optimal electromagnetic mechanical coupling coefficient μk, (c) optimal electrical tuning ratio fe, and (d) optimal electrical damping ratio ζe of approximate solutions.
(a)
(b)
(c)
(d)


### 4.1.3. Comparing Performances of Different Systems

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of five types of SDOF systems are compared under the same mass ratio of 0.02. From Figure 4, it is clear that the ERS-TMDI is superior to mitigate the vibration of the primary structure nearly across the whole frequency spectrum for force excitation system, as compared to that of classical TMD, ERS-TMDs, and system without a TMD, and the effective frequency band is also further widened. At their own resonant frequencies, the peak value of the normalized displacement Xn in the ERS-TMDI system is reduced by around 35% compared to that of the classical TMD, 18% to that of the ERS-TMDs, and 17% to that of series-type double-mass TMDs, while in the normalized displacement Xr, the relative reduction ratio to classical TMD and ERS-TMDs is 36% and 33%.Figure 4
Schematic of the optimal frequency responses for types of SDOF system, where the mass ratioμ=0.02. (a) The frequency response of the deformation of the primary structure and (b) the relative deformation of different buildings.
(a)
(b)


### 4.1.4. Comparing Performances of the ERS-TMDI under Different Inertance Ratios

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of the ERS-TMDI are compared under different inertance ratios. From Figure 5, H2 norm of the ERS-TMDI is better at the highest inertance ratio.Figure 5
Schematic of the optimal frequency responses under different inertance ratio: (a) the deformation of the primary structure and (b) the relative deformation of the ERS-TMDI.
(a)
(b)


## 4.1.1. Comparing ofH2 Tuning Laws of Exact and Approximate Solutions

Figure2 graphically depicts H2 tuning laws for the vibration mitigation when the primary structure is subjected to the force acceleration. It is obvious to see that the error between the exact solution and the approximate solution for the force excitation system is extremely small. Therefore, the approximate solution is a good alternative to avoid computational complexities in practice.Figure 2
Graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT; (b) optimal electromagnetic mechanical coupling coefficient μk; (c) optimal electrical tuning ratio fe; (d) optimal electrical damping ratio ζe of exact and approximate solutions under different mass ratios.
(a)
(b)
(c)
(d)


## 4.1.2. Three-Dimensional Graphical Representations ofH2 Tuning Laws for Approximate Solutions

Figure3 graphically depicts H2 tuning laws of approximate solutions in a three-dimensional way for the vibration mitigation when the primary structure is subjected to the force excitations.Figure 3
Three-dimensional graphical representations ofH2 tuning laws. (a) Optimal mechanical tuning ratio fT, (b) optimal electromagnetic mechanical coupling coefficient μk, (c) optimal electrical tuning ratio fe, and (d) optimal electrical damping ratio ζe of approximate solutions.
(a)
(b)
(c)
(d)


## 4.1.3. Comparing Performances of Different Systems

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of five types of SDOF systems are compared under the same mass ratio of 0.02. From Figure 4, it is clear that the ERS-TMDI is superior to mitigate the vibration of the primary structure nearly across the whole frequency spectrum for force excitation system, as compared to that of classical TMD, ERS-TMDs, and system without a TMD, and the effective frequency band is also further widened. At their own resonant frequencies, the peak value of the normalized displacement Xn in the ERS-TMDI system is reduced by around 35% compared to that of the classical TMD, 18% to that of the ERS-TMDs, and 17% to that of series-type double-mass TMDs, while in the normalized displacement Xr, the relative reduction ratio to classical TMD and ERS-TMDs is 36% and 33%.Figure 4
Schematic of the optimal frequency responses for types of SDOF system, where the mass ratioμ=0.02. (a) The frequency response of the deformation of the primary structure and (b) the relative deformation of different buildings.
(a)
(b)


## 4.1.4. Comparing Performances of the ERS-TMDI under Different Inertance Ratios

H
2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of the ERS-TMDI are compared under different inertance ratios. From Figure 5, H2 norm of the ERS-TMDI is better at the highest inertance ratio.Figure 5
Schematic of the optimal frequency responses under different inertance ratio: (a) the deformation of the primary structure and (b) the relative deformation of the ERS-TMDI.
(a)
(b)


## 4.2. Time-Domain Analyses of the ERS-TMDI

In this section, the Taipei 101 tower will be taken as a case study to be computed for simulated wind in order to demonstrate the dual effect of the ERS-TMDI. Results are presented in comparison with the classical TMD.


### 4.2.1. Parameters of Simulation

In this section we take the Taipei 101 tower as a case study and illustrate the dual functions of ERS-TMDI. Results are presented in comparison with the classical TMD.Taipei 101 is one of the tallest buildings in the world (449.2 m to roof and 509.2 m to spire). A TMD of 660 tonnes is suspended on the top of the building from the 92nd to the 87th floor to suppress the wind induced vibration [43]. The TMD is 0.78% of the modal mass, the first natural frequency is 0.146 Hz, and inherent damping of the building is 1%. In the case study, the parameters of the classical TMD are designed using H2 optimization as shown Table 1.Table 1
The parameters of simulation.
Description
Symbol
Value
Primary structure mass
m
s
8.46× 107 [kg]
Mechanical TMD mass
m
T
6.60× 105 [kg]
Inertance mass
b
3.30× 105 [kg]
Stiffness of the primary structure
k
s
7.12× 107 [N/m]
Stiffness of the mechanical TMD
k
T
8.21× 105 [N/m]
Total inductance of electrical resonator
L
1.17 [H]
Total capacitance of electrical resonator
C
1.02 [F]
Internal resistance of linear motor
R
i
0.1 [Ω]
External resistance
R
e
0.1 [Ω]
The constant of the electromagnetic transducer
k
v, kf
150


### 4.2.2. Wind Simulation

When the wind effect is analyzed, the Davenport wind spectrum expressed by (25a), (25b), and (25c) is taken as the target wind spectrum of wind velocity field [44]. The inverse fast Fourier transform method is used to generate the random fluctuating wind time series [45]. (25a)Sω=8πn2u∗2ω1+n24/3,(25b)n=600ωπV10,(25c)u∗=VzKln⁡z/z0,where Sω is the power spectrum of velocity fluctuation; ω is the frequency of the fluctuating wind; K is the terrain roughness factor of this area, which is 0.03; V10 is the mean wind speed, in m/s, at 10 m. It is taken as 39.93 m/s, which means that 50-year return period brings averaged over 10 minutes at a height of 10 m [46]. u∗ is the shear velocity of the flow, K=0.4. z is the height above the surface, z0 is the roughness length, and Vz is the mean wind speed of the height z.Figure6(a) is the typical fluctuating wind time history generated, whose duration is 1000 s and the time interval is 0.1 s. As shown in Figure 6(b), the generated wind time history is converted to power spectral density by IFFT, which is compared with the target Davenport spectrum. It is shown that the simulated spectrum is in agreement with Davenport spectrum well in most frequency bands.Figure 6
Simulated fluctuating wind time history: (a) wind speed and (b) power spectrum.
(a)
(b)With the information above, one can calculate fluctuating wind force by the following expression:(26)Ft=ρ·CD·A·V10·ut,where ρ=1.28 kg/m3 is the air density, CD=1.2 is the drag coefficient, the tributary area assuming is A=1600 m2, and ut is the fluctuating wind speed.


### 4.2.3. The Performance of Vibration Mitigation

In the time-history analyses, the characteristics of the model are identical to those used in the frequency-domain analyses described in Section4.1.From Figure7, it is clear that the ERS-TMDI sufficiently reduces the displacement of the primary structure. In addition, the performance of the vibration mitigation of the ERS-TMDI is clearly superior to that of the classical TMD. The peak value of the displacement of the primary structure specially is reduced by around 9.78% compared to that of the classical TMD, while, in accelerations, the relative reduction ratio is 6.95%. On the other hand, in the root mean square value of the primary structure displacement, the relative reduction is 9.71% and 11.49%.Figure 7
Time-history responses of (a) displacements and (b) accelerations of the primary structure: comparison between classical TMD and ERS-TMDI.
(a)
(b)


### 4.2.4. The Performance of Energy Harvesting

From Figure8 and Table 2, it is clear that the ERS-TMDI can also adequately harvest the power, whose peak power, average power, and total energy harvested are 1.34 × 103 w, 132.32 w, and 1.39 × 105 J, respectively.Table 2
The energy-related indicator.
Peak power/w
Average power/w
Energy harvested /J
1.34 × 103
132.32
1.39 × 105Figure 8
Time histories of the harvested power of ERS-TMDI.


## 4.2.1. Parameters of Simulation

In this section we take the Taipei 101 tower as a case study and illustrate the dual functions of ERS-TMDI. Results are presented in comparison with the classical TMD.Taipei 101 is one of the tallest buildings in the world (449.2 m to roof and 509.2 m to spire). A TMD of 660 tonnes is suspended on the top of the building from the 92nd to the 87th floor to suppress the wind induced vibration [43]. The TMD is 0.78% of the modal mass, the first natural frequency is 0.146 Hz, and inherent damping of the building is 1%. In the case study, the parameters of the classical TMD are designed using H2 optimization as shown Table 1.Table 1
The parameters of simulation.
Description
Symbol
Value
Primary structure mass
m
s
8.46× 107 [kg]
Mechanical TMD mass
m
T
6.60× 105 [kg]
Inertance mass
b
3.30× 105 [kg]
Stiffness of the primary structure
k
s
7.12× 107 [N/m]
Stiffness of the mechanical TMD
k
T
8.21× 105 [N/m]
Total inductance of electrical resonator
L
1.17 [H]
Total capacitance of electrical resonator
C
1.02 [F]
Internal resistance of linear motor
R
i
0.1 [Ω]
External resistance
R
e
0.1 [Ω]
The constant of the electromagnetic transducer
k
v, kf
150


## 4.2.2. Wind Simulation

When the wind effect is analyzed, the Davenport wind spectrum expressed by (25a), (25b), and (25c) is taken as the target wind spectrum of wind velocity field [44]. The inverse fast Fourier transform method is used to generate the random fluctuating wind time series [45]. (25a)Sω=8πn2u∗2ω1+n24/3,(25b)n=600ωπV10,(25c)u∗=VzKln⁡z/z0,where Sω is the power spectrum of velocity fluctuation; ω is the frequency of the fluctuating wind; K is the terrain roughness factor of this area, which is 0.03; V10 is the mean wind speed, in m/s, at 10 m. It is taken as 39.93 m/s, which means that 50-year return period brings averaged over 10 minutes at a height of 10 m [46]. u∗ is the shear velocity of the flow, K=0.4. z is the height above the surface, z0 is the roughness length, and Vz is the mean wind speed of the height z.Figure6(a) is the typical fluctuating wind time history generated, whose duration is 1000 s and the time interval is 0.1 s. As shown in Figure 6(b), the generated wind time history is converted to power spectral density by IFFT, which is compared with the target Davenport spectrum. It is shown that the simulated spectrum is in agreement with Davenport spectrum well in most frequency bands.Figure 6
Simulated fluctuating wind time history: (a) wind speed and (b) power spectrum.
(a)
(b)With the information above, one can calculate fluctuating wind force by the following expression:(26)Ft=ρ·CD·A·V10·ut,where ρ=1.28 kg/m3 is the air density, CD=1.2 is the drag coefficient, the tributary area assuming is A=1600 m2, and ut is the fluctuating wind speed.


## 4.2.3. The Performance of Vibration Mitigation

In the time-history analyses, the characteristics of the model are identical to those used in the frequency-domain analyses described in Section4.1.From Figure7, it is clear that the ERS-TMDI sufficiently reduces the displacement of the primary structure. In addition, the performance of the vibration mitigation of the ERS-TMDI is clearly superior to that of the classical TMD. The peak value of the displacement of the primary structure specially is reduced by around 9.78% compared to that of the classical TMD, while, in accelerations, the relative reduction ratio is 6.95%. On the other hand, in the root mean square value of the primary structure displacement, the relative reduction is 9.71% and 11.49%.Figure 7
Time-history responses of (a) displacements and (b) accelerations of the primary structure: comparison between classical TMD and ERS-TMDI.
(a)
(b)


## 4.2.4. The Performance of Energy Harvesting

From Figure8 and Table 2, it is clear that the ERS-TMDI can also adequately harvest the power, whose peak power, average power, and total energy harvested are 1.34 × 103 w, 132.32 w, and 1.39 × 105 J, respectively.Table 2
The energy-related indicator.
Peak power/w
Average power/w
Energy harvested /J
1.34 × 103
132.32
1.39 × 105Figure 8
Time histories of the harvested power of ERS-TMDI.


## 5. Conclusions

This paper investigates electromagnetic resonant shunt tuned mass-damper-inerter (ERS-TMDI) system, which consists of an auxiliary mechanical mass, an inertance mass, a spring, and an electromagnetic transducer shunted with an electrical RLC resonator.H2 tuning laws for the ERS-TMDI system are derived when the primary structure is subjected to wind excitations. Later on, the numerical analyses are conducted to illustrate the performance of vibration mitigation and energy harvesting.From the frequency-domain analyses, it is shown that the ERS-TMDI can enhance performance in terms of both vibration control and energy harvesting due to tuning the resonances of the mechanical shock absorber, the inertance, and the electrical resonator, as compared to classical TMD, in which only the mechanical shock absorber is tuned. Particularly, the ERS-TMDI improves the vibration mitigation by reducing the resonant peak by 35% compared to that of the classical TMD, 18% compared to that of the ERS-TMDs, and 17% compared to that of the series-type double-mass TMDs, while, in the relative displacement, the reduction ratio to the classical TMD and ERS-TMDs is 36% and 33%. In addition,H2 performances of vibration mitigation of the deformation of the primary structure and the relative deformation of the ERS-TMDI are better at the higher inertance ratio.From the time-domain analyses, it is shown that the ERS-TMDI effectively reduces the peak and root mean square of the displacement, which is superior to the classical TMD. In addition, the ERS-TMDI can also harvest the power efficiently, with a peak power of 1.34 × 103 w, an average power of 132.32 w, and a total power of 1.39 × 105 J.



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*Source: 4180134-2017-07-13.xml*