Wind Induced Vibration Control and Energy Harvesting of ... · ResearchArticle Wind Induced Vibration Control and Energy Harvesting of Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter

Research ArticleWind Induced Vibration Control and Energy Harvesting ofElectromagnetic Resonant Shunt Tuned Mass-Damper-Inerterfor Building Structures

Yifan Luo1 Hongxin Sun12 XiuyongWang1 Lei Zuo2 and Ning Chen1

1School of Civil Engineering Hunan University of Science and Technology Xiangtan 411201 China2Department of Mechanical Engineering Virginia Tech Blacksburg VA USA

Correspondence should be addressed to Hongxin Sun cehxsunhnusteducn

Received 20 February 2017 Accepted 24 April 2017 Published 13 July 2017

Academic Editor Sundararajan Natarajan

Copyright copy 2017 Yifan Luo et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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 offreedom system are introduced1198672 criteria performances of the ERS-TMDI are introduced in comparison with the classical tunedmass-damper (TMD) the electromagnetic resonant shunt series TMDs (ERS-TMDs) and series-type double-mass TMDs with theaim to minimize structure damage and simultaneously harvest energy under random wind excitation The closed form solutionsincluding the mechanical tuning ratio the electrical damping ratio the electrical tuning ratio and the electromagnetic mechanicalcoupling coefficient are obtained It is shown that the ERS-TMDI is superior to the classical TMD ERS-TMDs and series-typedouble-mass TMDs systems for protection from structure damage Meanwhile in the time domain a case study of Taipei 101 toweris presented to demonstrate the dual functions of vibration suppression and energy harvesting based on the simulation fluctuatingwind series which is generated by the inverse fast Fourier transform method The effectiveness and robustness of ERS-TMDI inthe frequency and time domain are illustrated

1 Introduction

It is a well-known fact that protecting structures againstdynamic loads such as winds and earthquakes has been oneof the chief purposes in structural design Many research hasbeen conducted to search for complementary energy dissipa-tion devices to suppress the harmful oscillation Among thesemethods the classical tuned mass-damper (TMD) [1] asdynamic vibration absorber has been proved to be one of themost effective and popular supplemental energy dissipationdevices [2ndash5] with many pragmatic installations [6ndash8]

Later on various methods to optimize the parameters ofTMD have been developed For the TMD with undampedprimary systems there are some common methods such asfixed-point method1198672 norm and119867infin optimal method Forthe TMD with damped primary systems substantive designmethods and tuning criteria have been raised [9] and the

TMD applied in nonlinear and distributed primary systemshas been studied [10 11]

Meanwhile energy harvesting from large-amplitude low-frequency oscillating primary structures has emerged as apromising research area [12 13] especially from TMD [14ndash19] It depends on replacing the energy-dissipating element ofthe TMD or supplementing with electromagnetic harvesterfor relatively large-scale applications [14ndash18] or piezoelec-tric materials even for relatively small-scale applications[17 19] Under this background a concept of tuned mass-damperharvester (TMDH) is presented in [20] in whichthe basic conversion consists of a linear voice coil motorconnected to a resistance emulator consisting of rectificationand variable impedance unit Along a similar path an electro-magnetic transducer connected with an energy harvesting-enabled circuit is used to transform vibration energy intoelectric power and also to supply controlled force vibrations

2 Shock and Vibration

mitigation for TMD-equipped multilayer buildings struc-tures in [11] In addition Ali and Adhikari proposed theidea of an energy harvester dynamic vibration absorber (EH-DVA) [19] in which electric energy is generated from strainsdeveloped in layers of piezoelectric material with the electriccircuit mounted onto the attached vibrating mass of a TMDThey also provided an analytical derivation of the EH-DVAunder harmonic and random excitation [19 21]

Lately an inerter a two-terminal mechanical device isproposed by Smith et al [22 23] The inerter is a devicethat provides a force proportional to the relative accelerationbetween its two terminals Moreover the inerter has beenstudied for improvement of car suspensions [24ndash26] andrailway vehicles [27 28] Other applications of inerters canalso be found in civil engineering for improving the per-formance of vibration control and displacement mitigationin base isolation systems It was proposed the idea of usingan inerter-like ball-screw mechanism to be employed in aviscous mass-damper or tuned viscous mass-damper systemin [29] A type of electromagnetic inertial mass-damperusing a ball-screw mechanism and a motor was presented tocontrol the vibration of structures subjected to earthquakesin [30] Moreover an inertia-based passive control basedon a gyro-mass-damper using gear assemblies was alsostudied as a type of supplemental damper [31] and as abase isolation system [32] Lazar et al [33 34] proposed aninerter-based system with a configuration similar to that ofa TMD which is termed as tuned inerter damper (TID)and developed an analytical tuning rule of a TID based onthe Den Hartogrsquos method Marian and Giaralis [35] alsoproposed the concept of tunedmass-damper-inerter (TMDI)to mitigate the oscillatory motion of support systems whichare stochastically excited They also developed the optimalTMDI parameters for undamped primary structures underwhite noise excitation in closed-form as functions of theTMD mass and the inertance Further they proposed theenergy harvesting-enabled tuned mass-damper-inerter (EH-TMDI) configuration for simultaneous vibration suppressionand energy harvesting [36]

Inspired by the idea of EH-TMDI [35 36] this paperproposes a novel inerter-based dynamic vibration absorbercalled electromagnetic resonant shunt tuned mass-damper-inerter (ERS-TMDI) in which we replace the dissipatedelement of the TMDI with the electromagnetic transducershunted with a resonant R-L-C circuit for simultaneousvibration control and energy harvesting In addition thispaper derives the closed-form solution of optimal designparameters to the ERS-TMDI for building structures underrandom wind and presented numerical analyses 1198672 normoptimal method is employed for the undamped single degreeof freedom (SDOF) structure with the objective to minimizethe mean squared value of the displacement of the primarystructureThe advantages of the ERS-TMDIwill be illustratedthrough the comparison with the classical TMD series-typedouble-mass TMDs [37] and ERS-TMDs [38]

The remainder of this paper is organized as followsIn Section 2 the governing equations of the SDOF systemwith ERS-TMDI are derived Section 3 derives optimal 1198672closed-form solutions to the ERS-TMDI systems under the

wind force excitation In Section 4 the numerical analysis ofstructures is presented in comparison with the classical TMDand ERS-TMDs Conclusions are drawn in Section 5

2 The Single Degree of Freedom SystemModel of the Electromagnetic ResonantShunt Tuned Mass-Damper-Inerter

As is shown in Figure 1(a) for easily understanding thereduction effects of the damper a multistory building issimplified as a general multidegree of freedom (MDOF)system Further Figure 1(a) also shows the configuration ofERS-TMDI for the MDOF where the ERS-TMDI is obtainedby replacing the original energy dissipative damping 119888119879 inclassical TMD with an electromagnetic transducer shuntedwith a resistance 119877 capacitance 119862 and inductance 119871 [16 39]and an inertance b a two-terminal mechanical device isassigned between the absorber and next story of structure forsimultaneous vibration control and energy harvesting underthe force excitation of the primary structure

When multistory buildings are subjected to wind vibra-tion the first-order modal played a very important role inthe analysis of vibration Therefore the primary structure ischaracterized as single degree of freedom (SDOF) structurewith a linear spring of stiffness 119896119904 a mass119898119904 and a dampingcoefficient 119888119904 as shown in Figure 1(b)

The motion equations of the combined SDOF and ERS-TMDI system shown in Figure 1(b) can be given by

119898119904119904 + 119888119904119904 + 119896119904119909119904 minus 119896119879 (119909119879 minus 119909119904) + 119896119891119868 = 119865(119898119879 + 119887) 119879 + 119896119879 (119909119879 minus 119909119904) minus 119896119891119868 = 0119896V ( 119909119879 minus 119909119904) + 119877119868 + 119871 119868 + 1119862 int 119868 119889119905 = 0

(1)

when the primary structure is undamped the above can besimplified in the Laplace domain

1198981199041199091199041199042+119896119904119909119904 minus 119896119879 (119909119879 minus 119909119904) + 119896119891119868 = 119865(119898119879 + 119887) 1199091198791199042 + 119896119879 (119909119879 minus 119909119904) minus 119896119891119868 = 0

119896V (119909119879 minus 119909119904) 119904 + 119877119868 + 119871119868119904 + 119868119862119904 = 0(2)

normalizing the frequency and setting 119904 = 119895120596 we can have

119909119904 (119895120572)2 + 119909119904 minus 1205831198912119879 (119909119879 minus 119909119904) + 119896119891119896119904 119868 =119865119896119904

(1 + 120575) 119909119879 (119895120572)2 + 1198912119879 (119909119879 minus 119909119904) minus 119896119891120583119896119904 119868 = 0119896V (119909119879 minus 119909119904) (119895120572) + 2120577119890119891119890119871119868 + 119871119868 (119895120572) + 1198681198711198912119890 1(119895120572)= 0

(3)

Shock and Vibration 3

(b)

F

F

F

F

F

kncn

Cnbn

kTn

mTn

In

Ln

Rn

enEMFfn

EMF

mn

knminus1 cnminus1

Cnminus1bnminus1

kTnminus1

Inminus1

Lnminus1

Rnminus1

enminus1EMFfnminus1

EMF

mnminus2

k1c1

C1b1

kT1

m1

I1

L1

R1

e1EMFf1

EMFmT1

kscs

Cb

mT

L

R

eEMFfEMF

ms

I

kT

(a)

mnminus1

mTnminus1

Figure 1 Schematic of the coupled structure and tuning damper system (a) a general MDOF and (b) the SDOF structure-damper system

rewriting relative displacement 119909119903 = 119909119879 minus 119909119904119909119904 ((119895120572)2 + 1) minus 1205831198912119879119909119903 + 119896119891119896119904 119868 =

119865119896119904 119909119903 ((1 + 120575) (119895120572)2 + 1198912119879) + 119909119904 (1 + 120575) (119895120572)2 minus 119896119891120583119896119904 119868 = 0119896V119909119903 (119895120572) + 119871119868(2120577119890119891119890 + (119895120572) + 1198912119890 1(119895120572)) = 0

(4)

where the nature frequency of the primary structure 120596119904 =radic119896119904119898119904 the nature frequency of the absorber 120596119879 = radic119896119879119898119879the resonant frequency of the circuit 120596119890 = 1radic119871119862 the massratio of the absorber to the primary structure 120583 = 119898119879119898119904the mass ratio of inertance to the primary structure 120575 =119887119898119879 the mechanical tuning ratio 119891119879 = 120596119879120596119904 the electricaltuning ratio 119891119890 = 120596119890120596119904 the normalized frequency 120572 =120596120596119904 and the electrical damping ratio of the circuit 120577119890 =119877(2119871120596119890) where the total resistance 119877 = 119877119894 + 119877119890 and 119877119890 isthe equivalent external load the electromagnetic mechanicalcoupling coefficient 120583119896 = 119896V119896119891(119896119879119871) where 119896V and 119896119891 are thevoltage constant and the force constant of the electromagnetictransducer

3 1198672 Optimization for the ERS-TMDI

31 Vibration Mitigation of the ERS-TMDI If the systemsuffered from the wind force rather than the sinusoidalexcitation 1198672 optimization is more desirable for evaluatingthe system performance for it is the root mean square (RMS)value of the performance under unit Gaussian white noiseinput This is a good approximation when the frequency of

the wind force is broad when compared with the naturalfrequency of the building For mitigating the vibration of theprimary structure when subjected to the excitation force 1198651198672 norm is minimized from 119865119896119904 to the deformation of theprimary structure and 119909119904 is defined as

PIV = 119864 [1199092119904 ]21205871205961199041198780 =

⟨1199092119904⟩21205871205961199041198780 (5)

where the symbols 119864[sdot] and ⟨sdot⟩ refer to the ensemble andtemporal averages respectively and 1198780 is the uniform powerspectrum function The RMS value of the deformation of theSDOF structure mass 119909119904 can be obtained as

⟨1199092119904⟩ = 1205961199041198780 intinfinminusinfin

100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (6)

where 119883119899 in (6) is the norm of the frequency responsefunction from 119865119896119904 to the deformation of the primarystructure 119909119904 and 119895 = radicminus1 is the unit imaginary numberSubstituting (6) into (5) the performance index PIV in (5) canbe expressed as

PIV = 12120587 intinfin

minusinfin

100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (7)

In addition the normalized frequency response function 119883119899from 119865119896119904 to 119909119904 can be written in the dimensionless form byusing the aforementioned dimensionless parameters whichis


119883119899 = 119909119904119865119896119904 =1198912119890 1198912119879 + 21198911198901198912119879120577119890 (119895120572) + (1198912119890 120595 + 1198912119879 (1 + 120583119896)) (119895120572)2 + 2119891119890120595120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6

1198600 = 1198912119890 1198912119879 1198601 = 21198912119879119891119890120577119890 1198602 = 1198912119890 120595 + 1198912119879 (1 + 120583119896) + 1198912119890 1198912119879 (1 + 120583120595) 1198603 = 2119891119890120577119890 (120595 + 1198912119879 (1 + 120583120595)) 1198604 = 1198912119879 (1 + 120583119896) (1 + 120583120595) + (1 + 1198912119890 ) 120595 1198605 = 2119891119890120577119890120595 1198606 = 120595(8)

where 120595 = 1 + 120575The integral in (7) can be solved using the general formula

in [40] Hence the performance index in (7) can be obtainedas a function of the four design parameters 119891119879 120583119896 119891119890 120577119890 andthe given parameters 120583 120595PIV = 1411989111989011989121198791205831198961205771198901205831205952 (120595

2 + 1198914119890 1205952 + 1198912119890 1205952 (minus2 + 41205772119890 )+ 1198912119879 (minus2 (1 + 120583119896) 120595 minus 21198914119890 120595 (1 + 120583120595) + 1198912119890 120595 (2+ 2 (1 + 120583119896) + 120583120595 + 120583 (1 + 120583119896) 120595 minus 81205772119890 minus 41205831205951205772119890 ))+ 1198914119879 ((1 + 120583119896)2 + 120583 (1 + 120583119896)2 120595 + 1198914119890 (1 + 3120583120595+ 312058321205952 + 12058331205953) + 1198912119890 (minus2 (1 + 120583119896) minus 4120583 (1 + 120583119896) 120595minus 21205832 (1 + 120583119896) 1205952 + 41205772119890 + 81205831205951205772119890 + 4120583212059521205772119890 )))

(9)

To minimize the performance index PIV regarding thevibration mitigation performance the derivatives of PIV withrespect to all the design parameters should be equal to zeroThus we have

120597PIV120597119891119879 = 0120597PIV120597120577119890 = 0120597PIV120597119891119890 = 0120597PIV120597120583119896 = 0

(10)

Therefore the following simultaneous gradientsrsquo equationscan be obtained from (10)

1198914119879 (1 + 120583119896)2 (1 + 120583120595) + 1198914119890 1198914119879 (1 + 120583120595)3minus 21198912119890 1198914119879 (1 + 120583120595)2 (1 + 120583119896 minus 21205772119890 )minus 1205952 (1 + 1198914119890 + 1198912119890 (minus2 + 41205772119890 )) = 0

(11a)

minus 21198912119879 (1 + 120583119896) 120595 + (1198914119879 (1 + 120583119896)2 minus 21198914119890 1198912119879120595)sdot (1 + 120583120595) + 1198914119890 1198914119879 (1 + 120583120595)3 minus 21198912119890 1198914119879 (1 + 120583120595)2sdot (1 + 120583119896 + 21205772119890 ) + 1198912119890 1198912119879120595 (2 + 120583120595) (2 + 120583119896 + 41205772119890 )+ 1205952 (1 + 1198914119890 minus 21198912119890 (1 + 21205772119890 )) = 0

(11b)

21198912119879 (1 + 120583119896) 120595 + (minus1198914119879 (1 + 120583119896)2 minus 61198914119890 1198912119879120595)sdot (1 + 120583120595) + 31198914119890 1198914119879 (1 + 120583120595)3 + 1198912119890 1198912119879120595 (2 + 120583120595)sdot (2 + 120583119896 minus 41205772119890 ) minus 21198912119890 1198914119879 (1 + 120583120595)2sdot (1 + 120583119896 minus 21205772119890 )+ 1205952 (minus1 + 31198914119890 + 1198912119890 (minus2 + 41205772119890 )) = 0

(11c)

minus 21198912119879120595 + (1198914119879 (1 minus 1205832119896) minus 21198914119890 1198912119879120595) (1 + 120583120595)+ 1198914119890 1198914119879 (1 + 120583120595)3 + 21198912119890 1198914119879 (1 + 120583120595)2 (minus1 + 21205772119890 )minus 21198912119890 1198912119879120595 (2 + 120583120595) (minus1 + 21205772119890 )+ 1205952 (1 + 1198914119890 + 1198912119890 (minus2 + 41205772119890 )) = 0

(11d)

Solving this set of equations is nontrivial as it involvesmultiple nonlinear high-order variables but in the followingwe will simply summarize the solving process and thendirectly present the final resultsThis will highlight the resultsand avoid prolixity By combining (11d) with the other threeequations in (11a) (11b) (11c) and (11d) we can eliminate 120577119890 toobtain a new equation set in design variables 119891119879 120583119896 119891119890 Thenusing similar manipulations we can also eliminate 119891119879 and 119891119890from the new equation set and obtain (11a) (11b) (11c) and(11d) in only variable 120583119896 Similarly1198672 norms optimal can beobtained by (11a) (11b) (11c) and (11d) as

119891opt119879 = radic120595 (119903 minus 120583120595)2 (1 + 120583120595) 120583opt119896= 4120583120595 (4 + 6120583120595 + 119903)16 + 19120583120595


119891opt119890 = radic32 + 2512058321205952 + 120583120595 (58 + 119903)2 (16 + 35120583120595 + 1912058321205952)

120577opt119890 = radic(1 (16 + 19120583120595) 120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)))radic16 + 23120583120595 + 812058321205952 (12)

where 119903 = radic16 + 32120583120595 + 1712058321205952 At 1198672 optimal tuning condition the performance indexPIoptV is

PIoptV

= minus(radic8 + 231205831205952 + 412058321205952 (78412058341205954 + 192 (4 + 119903) + 20120583120595 (156 + 31119903) + 12058331205953 (3147 + 232119903) + 12058321205952 (4716 + 661119903)))(2 (1 + 120583120595) (16 + 19120583120595) (120583120595 minus 119903) (4 + 6120583120595 + 119903)radic(32 + 2512058321205952 + 120583120595 (58 + 119903)) (16 + 35120583120595 + 1912058321205952)radic120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)) (16 + 19120583120595))

(13)

Because the mass ratio 120583 is a pretty small numberregularly less than 01 for building-TMD systems the squareroot term 119903 = radic16 + 32120583120595 + 1712058321205952 can be about as 4(1+120583120595)by segmentally ignoring the terms involving second powersin 120583120595 Hence a compendious approximate solution set canbe obtained as

119891optlowast119879 = radic120595 (4 + 3120583120595)2 (1 + 120583120595) 120583optlowast119896

= 4120583120595 (8 + 10120583120595)16 + 19120583120595

119891optlowast119890 = radic 32 + 2912058321205952 + 621205831205952 (16 + 35120583120595 + 1912058321205952) 120577optlowast119890= radic(1 (16 + 19120583120595) 120583120595 (14212058331205953 + 49512058321205952 + 544120583120595 + 192))radic16 + 23120583120595 + 812058321205952

(14)

with the approximate optimal performance index PIoptlowastV

PIoptlowastV

= radic16 + 23120583120595 + 812058321205952 (1536 + 6368120583120595 + 984012058321205952 + 671912058331205953 + 171212058341205954)4 (1 + 120583120595) (4 + 3120583120595) (4 + 5120583120595) (16 + 19120583120595)radic(64 + 124120583120595 + 5812058321205952) (16 + 35120583120595 + 1912058321205952)radic120583120595 (192 + 544120583120595 + 49512058321205952 + 14212058331205953) (16 + 19120583120595)

(15)

Therefore it is easy to obtain the corresponding optimalabsorber stiffness 119896opt119879 the inductance 119871opt the capacitance119862opt and the total resistance 119877opt with the above parameters

119871opt = 119896V119896119891120583opt119896119891opt119879

21198981198791205962119904 119896opt119879 = 119891opt

119879

21198981198791205962119904 119877opt = 2119896V119896119891120577opt119890 119891opt

119890

120583opt119896119891opt119879

2119898119879120596119904

119862opt = 119898119879120583opt119896 119891opt119879

2

119896V119896119891119891opt119890

2

(16)

Moreover there is the need to declare that1198672 tuning lawfor the classical TMD system is [41]

119891opt = 11 + 120583radic2 + 1205832

120577opt = radic 120583 (4 + 3120583)8 (1 + 120583) (2 + 120583)

(17)

Moreover the approximate1198672 tuning law for the ERS-TMDsis [42]

119891optlowastlowast119879 = radic4 + 31205832 (1 + 120583) 120583optlowastlowast119896

= 32120583 + 40120583216 + 19120583


119891optlowastlowast119890 = radic 32 + 62120583 + 2912058322 (16 + 35120583 + 191205832)

120577optlowastlowast119890 = radic120583 (192 + 544120583 + 495)256 minus 96120583 minus 271205832 (18)

And the optimal parameters of the series-type double-massTMDs [37] are

120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)

where 120583119861opt is the mass ratio of the further mass to the closermass ]opt is the undamped natural frequency ratio of thecloser mass to the primary system ]119861opt is the undampednatural frequency ratio of the furthermass to the closer mass1205772opt is the damping ratio of the closer mass 1205773opt is thedamping ratio of the closer mass

In addition the normalized frequency response function119883119903 from 119865119896119904 to the relative displacement of the primarystructure 119909119903 can also be expressed in the dimensionlessform by using the aforementioned dimensionless parameterswhich are

119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)

32 Energy Harvesting of the Electromagnetic Resonant ShuntTuned Mass-Damper-Inerter To optimize the energy har-vesting of the ERS-TMDI it is our wish to maximize theaverage electrical power on the external load 119877119890 The instantpower on the external load is

119875 = 1198771198901198682 (21)

The normalized frequency response function from 119865119896119904to 119868 can be written in the dimensionless form by using theaforementioned dimensionless parameters which are

119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)

similarly the normalized frequency response function from119865119896119904 toradic119875 can also be written

119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)

At the optimal 1198672 tuning condition the performance indexPI119890 is

PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)

4 Numerical Analyses

41 Frequency-Domain Analyses of the ERS-TMDI

411 Comparing of1198672 Tuning Laws of Exact and ApproximateSolutions Figure 2 graphically depicts1198672 tuning laws for thevibration mitigation when the primary structure is subjected

to the force acceleration It is obvious to see that the errorbetween the exact solution and the approximate solutionfor the force excitation system is extremely small Thereforethe approximate solution is a good alternative to avoidcomputational complexities in practice

412 Three-Dimensional Graphical Representations of 1198672Tuning Laws for Approximate Solutions Figure 3 graphicallydepicts 1198672 tuning laws of approximate solutions in a three-dimensional way for the vibration mitigation when theprimary structure is subjected to the force excitations

413 Comparing Performances of Different Systems 1198672 per-formances of vibration mitigation of the deformation of theprimary structure and the relative deformation of five typesof SDOF systems are compared under the same mass ratio of002 From Figure 4 it is clear that the ERS-TMDI is superiorto mitigate the vibration of the primary structure nearlyacross the whole frequency spectrum for force excitation


08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx

휇 = 01 of exact solution

solution

solution

solution(a)

00

01

02

03

04

05

00 02 04 06 08 10Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)

Figure 2 Graphical representations of 1198672 tuning laws (a) Optimal mechanical tuning ratio 119891119879 (b) optimal electromagnetic mechanicalcoupling coefficient 120583119896 (c) optimal electrical tuning ratio119891119890 (d) optimal electrical damping ratio 120577119890 of exact and approximate solutions underdifferent mass ratios

system as compared to that of classical TMD ERS-TMDsand system without a TMD and the effective frequency bandis also further widened At their own resonant frequenciesthe peak value of the normalized displacement 119883119899 in theERS-TMDI system is reduced by around 35 compared tothat of the classical TMD 18 to that of the ERS-TMDs and17 to that of series-type double-mass TMDs while in thenormalized displacement 119883119903 the relative reduction ratio toclassical TMD and ERS-TMDs is 36 and 33

414 Comparing Performances of the ERS-TMDI under Dif-ferent Inertance Ratios 1198672 performances of vibration miti-gation of the deformation of the primary structure and the

relative deformation of the ERS-TMDI are compared underdifferent inertance ratios FromFigure 51198672 normof the ERS-TMDI is better at the highest inertance ratio

42 Time-Domain Analyses of the ERS-TMDI In this sectionthe Taipei 101 tower will be taken as a case study to be com-puted for simulated wind in order to demonstrate the dualeffect of the ERS-TMDI Results are presented in comparisonwith the classical TMD

421 Parameters of Simulation In this section we take theTaipei 101 tower as a case study and illustrate the dual


0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT

Inertance ratio 훿 Mass ratio 휇

(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)

Figure 3 Three-dimensional graphical representations of 1198672 tuning laws (a) Optimal mechanical tuning ratio 119891119879 (b) optimalelectromagnetic mechanical coupling coefficient 120583119896 (c) optimal electrical tuning ratio 119891119890 and (d) optimal electrical damping ratio 120577119890 ofapproximate solutions

functions of ERS-TMDI Results are presented in comparisonwith the classical TMD

Taipei 101 is one of the tallest buildings in the world(4492m to roof and 5092m to spire) A TMD of 660 tonnesis suspended on the top of the building from the 92nd to the87th floor to suppress the wind induced vibration [43] TheTMD is 078 of the modal mass the first natural frequencyis 0146Hz and inherent damping of the building is 1 In thecase study the parameters of the classical TMD are designedusing1198672 optimization as shown Table 1

422 Wind Simulation When the wind effect is analyzedthe Davenport wind spectrum expressed by (25a) (25b) and(25c) is taken as the target wind spectrum of wind velocityfield [44] The inverse fast Fourier transform method is usedto generate the random fluctuating wind time series [45]

119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)

Table 1 The parameters of simulation

Description Symbol Value

Primary structure mass 119898119904 846 times 107 [kg]Mechanical TMDmass 119898119879 660 times 105 [kg]Inertance mass 119887 330 times 105 [kg]Stiffness of the primary structure 119896119904 712 times 107 [Nm]

Stiffness of the mechanical TMD 119896119879 821 times 105 [Nm]

Total inductance of electricalresonator

119871 117 [H]

Total capacitance of electricalresonator

119862 102 [F]

Internal resistance of linear motor 119877119894 01 [Ω]External resistance 119877119890 01 [Ω]The constant of theelectromagnetic transducer

119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs

Classical TMDDouble-mass TMDs

Normalized frequency 훼

ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs

Classical TMDERS-TMDI (훿 = 05)

07 08 09 1 11 12 13Normalized frequency 훼

Nor

mal

ized

disp

lace

men

tXr

(b)

Figure 4 Schematic of the optimal frequency responses for types of SDOF system where themass ratio 120583 = 002 (a)The frequency responseof the deformation of the primary structure and (b) the relative deformation of different buildings

07 08 09 1 11 12 13123456789

101112


ERS-TMDI (훿 = 01)ERS-TMDI (훿 = 05)ERS-TMDI (훿 = 1)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)

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

119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)

where 119878(120596) is the power spectrum of velocity fluctuation120596 is the frequency of the fluctuating wind 119870 is the terrainroughness factor of this area which is 003 11988110 is the meanwind speed in ms at 10m It is taken as 3993ms which

means that 50-year return period brings averaged over 10minutes at a height of 10m [46] 119906lowast is the shear velocity ofthe flow 119870 = 04 119911 is the height above the surface 1199110 is theroughness length and119881119911 is themeanwind speed of the height119911

Figure 6(a) is the typical fluctuating wind time historygenerated whose duration is 1000 s and the time intervalis 01 s As shown in Figure 6(b) the generated wind timehistory is converted to power spectral density by IFFT which


100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)

Davenport target specSimulated wind spec

1E minus 31E minus 3

Win

d sp

ectr

um (m

2s

3)

(b)

Figure 6 Simulated fluctuating wind time history (a) wind speed and (b) power spectrum

0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)

Without controlERS-TMDIClassical TMD

minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)

Figure 7 Time-history responses of (a) displacements and (b) accelerations of the primary structure comparison between classical TMDand ERS-TMDI

is compared with the target Davenport spectrum It is shownthat the simulated spectrum is in agreement with Davenportspectrum well in most frequency bands

With the information above one can calculate fluctuatingwind force by the following expression

119865 (119905) = 120588 sdot 119862119863 sdot 119860 sdot 11988110 sdot 119906 (119905) (26)

where 120588 = 128 kgm3 is the air density 119862119863 = 12 is the dragcoefficient the tributary area assuming is 119860 = 1600m2 and119906(119905) is the fluctuating wind speed

423 The Performance of Vibration Mitigation In the time-history analyses the characteristics of the model are identicalto those used in the frequency-domain analyses described inSection 41

From Figure 7 it is clear that the ERS-TMDI sufficientlyreduces the displacement of the primary structure In addi-tion the performance of the vibration mitigation of the ERS-TMDI is clearly superior to that of the classical TMD Thepeak value of the displacement of the primary structurespecially is reduced by around 978 compared to that of the


Table 2 The energy-related indicator

Peak powerw Average powerw Energy harvested J134 times 103 13232 139 times 105

0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)

Figure 8 Time histories of the harvested power of ERS-TMDI

classical TMD while in accelerations the relative reductionratio is 695 On the other hand in the root mean squarevalue of the primary structure displacement the relativereduction is 971 and 1149

424 The Performance of Energy Harvesting From Figure 8and Table 2 it is clear that the ERS-TMDI can also adequatelyharvest the power whose peak power average power andtotal energy harvested are 134 times 103 w 13232 w and 139 times105 J respectively

5 Conclusions

This paper investigates electromagnetic resonant shunt tunedmass-damper-inerter (ERS-TMDI) system which consists ofan auxiliary mechanical mass an inertance mass a springand an electromagnetic transducer shunted with an electricalRLC resonator 1198672 tuning laws for the ERS-TMDI systemare derived when the primary structure is subjected to windexcitations Later on the numerical analyses are conducted toillustrate the performance of vibration mitigation and energyharvesting

From the frequency-domain analyses it is shown thatthe ERS-TMDI can enhance performance in terms of bothvibration control and energy harvesting due to tuning theresonances of the mechanical shock absorber the inertanceand the electrical resonator as compared to classical TMD inwhich only the mechanical shock absorber is tuned Partic-ularly the ERS-TMDI improves the vibration mitigation byreducing the resonant peak by 35 compared to that of theclassical TMD 18 compared to that of the ERS-TMDs and17 compared to that of the series-type double-mass TMDswhile in the relative displacement the reduction ratio to the

classical TMD and ERS-TMDs is 36 and 33 In addition1198672 performances of vibration mitigation of the deformationof the primary structure and the relative deformation of theERS-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 ofthe displacement which is superior to the classical TMDIn addition the ERS-TMDI can also harvest the powerefficiently with a peak power of 134times 103 w an average powerof 13232 w and a total power of 139 times 105 JConflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by National Natural Science Foun-dation of China under Grant 51508185 National Key BasicResearch Scheme of China (973 program) under Grant2015CB057702 and Natural Science Foundation of HunanProvince of China under Grant 2015JJ3073 The authorHongxin Sun is also grateful for the financial support fromChina Scholarship Council to conduct research as a visitingscholar in Lei Zuorsquos Lab at Virginia Tech

References

[1] J P D Hartog Mechanical vibration 1956[2] R J Mcnamara ldquoTunedmass dampers for buildingsrdquo Journal of

the Structural Division vol 1 pp 1785ndash1798 1977[3] J C H Chang and T T Soong ldquoStructural control using

active tunedmass damperrdquo Journal of the EngineeringMechanicsDivision vol 106 no 306 pp 306ndash311 1980

[4] A M Kaynia D Veneziano and J M Biggs ldquoSeismic effective-ness of tuned mass dampersrdquo Journal of the Structural Divisionvol 107 pp 1465ndash1484 1981

[5] J S Bae J H Hwang D G Kwag J Park and D J InmanldquoVibration suppression of a large beam structure using tunedmass damper and eddy current dampingrdquo Shock amp Vibration10 pages 2014

[6] H Frahm Device for Damping Vibrations of Bodies GooglePatents 1911

[7] P Irwin J Kilpatrick J Robinson and A Frisque ldquoWind andtall buildings Negatives and positivesrdquo Structural Design of Tallamp Special Buildings vol 17 pp 915ndash928 2008

[8] J Salvi and E Rizzi ldquoClosed-form optimum tuning formulasfor passive tuned mass dampers under benhcmark excitationsrdquoSmart Structures amp Systems vol 17 pp 231ndash256 2016

[9] A Ghosh and B Basu ldquoA closed-form optimal tuning criterionfor tmd in damped structuresrdquo Structural Control amp HealthMonitoring vol 14 pp 681ndash692 2007

[10] Y L Cheung and W O Wong ldquoHinfin and H2 optimizationsof a dynamic vibration absorber for suppressing vibrations inplatesrdquo Journal of Sound amp Vibration vol 320 pp 29ndash42 2009

[11] M H Miguelez L Rubio J A Loya and J Fernandez-SaezldquoImprovement of chatter stability in boring operations withpassive vibration absorbersrdquo International Journal ofMechanicalSciences vol 52 no 10 pp 1376ndash1384 2010


[12] S Zhu W-A Shen and Y-L Xu ldquoLinear electromagneticdevices for vibration damping and energy harvestingModelingand testingrdquo Engineering Structures vol 34 pp 198ndash212 2012

[13] J Zhu and W Zhang ldquoCoupled analysis of multi-impactenergy harvesting from low-frequency wind induced vibra-tionsrdquo Smart Materials and Structures vol 24 no 4 Article ID045007 2015

[14] I L Cassidy J T Scruggs S Behrens and H P Gavin ldquoDesignand experimental characterization of an electromagnetic trans-ducer for large-scale vibratory energy harvesting applicationsrdquoJournal of IntelligentMaterial Systems and Structures vol 22 no17 pp 2009ndash2024 2011

[15] W A Shen S Zhu and Y L Xu ldquoAn experimental study onself-powered vibration control and monitoring system usingelectromagnetic tmd and wireless sensorsrdquo Sensors amp ActuatorsA Physical vol 180 pp 166ndash176 2012

[16] X Tang and L Zuo ldquoSimultaneous energy harvesting andvibration control of structures with tuned mass dampersrdquoJournal of Intelligent Material Systems amp Structures vol 23 pp2117ndash2127 2012

[17] L Zuo and X Tang ldquoLarge-scale vibration energy harvestingrdquoJournal of Intelligent Material Systems amp Structures vol 24 pp1405ndash1430 2013

[18] A Gonzalez-Buelga L R Clare A Cammarano S A NeildS G Burrow and D J Inman ldquoAn optimised tuned massdamperharvester devicerdquo Structural Control ampHealthMonitor-ing vol 21 pp 1154ndash1169 2014

[19] S F Ali and S Adhikari ldquoEnergy harvesting dynamic vibrationabsorbersrdquo Journal of Applied Mechanics Transactions ASMEvol 80 no 4 Article ID 041004 2013

[20] A Gonzalez-Buelga L R Clare A Cammarano S A NeildS G Burrow and D J Inman ldquoAn optimised tuned massdamperharvester devicerdquo Structural Control and Health Moni-toring vol 21 no 8 pp 1154ndash1169 2014

[21] C Madhav and S F Ali ldquoHarvesting energy from vibrationabsorber under random excitationsrdquo IFAC-PapersOnLine vol49 no 1 pp 807ndash812 2016

[22] M C Smith ldquoSynthesis of mechanical networks the inerterrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 47 no 10 pp 1648ndash1662 2002

[23] M Z Q Chen C Papageorgiou F Scheibe F-CWang andMSmith ldquoThe missing mechanical circuit elementrdquo IEEE Circuitsand Systems Magazine vol 9 no 1 pp 10ndash26 2009

[24] M C Smith and F U-C Wang ldquoPerformance benefits inpassive vehicle suspensions employing inertersrdquo Vehicle SystemDynamics vol 42 no 4 pp 235ndash257 2004

[25] C Papageorgiou and M C Smith ldquoPositive real synthesisusing matrix inequalities for mechanical networks Applicationto vehicle suspensionrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 3 pp 423ndash435 2006

[26] Y LHuMZQChen andZ Shu ldquoPassive vehicle suspensionsemploying inerters with multiple performance requirementsrdquoJournal of Sound and Vibration vol 333 no 8 pp 2212ndash22252014

[27] J Z Jiang A Z Matamoros-Sanchez R M Goodall andM CSmith ldquoPassive suspensions incorporating inerters for railwayvehiclesrdquo Vehicle System Dynamics vol 50 no 1 pp 263ndash2762012

[28] F-C Wang M-K Liao B-H Liao W-J Su and H-A ChanldquoThe performance improvements of train suspension systems

with mechanical networks employing inertersrdquo Vehicle SystemDynamics vol 47 no 7 pp 805ndash830 2009

[29] K Ikago K Saito and N Inoue ldquoSeismic control of single-degree-of-freedom structure using tuned viscousmass damperrdquoEarthquake Engineeringamp Structural Dynamics vol 41 pp 453ndash474 2012

[30] Y Nakamura A Fukukita K Tamura et al ldquoSeismic responsecontrol using electromagnetic inertial mass dampersrdquo Earth-quake Engineering amp Structural Dynamics vol 43 pp 507ndash5272014

[31] R Mirza Hessabi and OMercan ldquoInvestigations of the applica-tion of gyro-mass dampers with various types of supplementaldampers for vibration control of building structuresrdquo Engineer-ing Structures vol 126 pp 174ndash186 2016

[32] M Saitoh ldquoOn the performance of gyro-mass devices fordisplacement mitigation in base isolation systemsrdquo StructuralControl and Health Monitoring vol 19 no 2 pp 246ndash259 2012

[33] I F Lazar S A Neild and D J Wagg ldquoUsing an inerter-based device for structural vibration suppressionrdquo EarthquakeEngineering amp Structural Dynamics vol 43 pp 1129ndash1147 2014

[34] I F Lazar S A Neild and D J Wagg ldquoVibration suppressionof cables using tuned inerter dampersrdquo Engineering Structuresvol 122 pp 62ndash71 2016

[35] L Marian and A Giaralis ldquoOptimal design of a novel tunedmass-damper-inerter (TMDI) passive vibration control config-uration for stochastically support-excited structural systemsrdquoProbabilistic Engineering Mechanics vol 38 pp 156ndash164 2014

[36] J Salvi and A Giaralis ldquoConcept study of a novel energyharvesting-enabled tuned mass-damper-inerter (eh-tmdi)device for vibration control of harmonically-excited structuresrdquoJournal of Physics Conference Series vol 744 Article ID 0120822016

[37] T Asami Optimal design of double-mass dynamic vibrationabsorbers arranged in series or in parallel 2016

[38] Y Liu C-C Lin J Parker and L Zuo ldquoExact H2 Optimal Tun-ing and Experimental Verification of Energy-Harvesting SeriesElectromagnetic Tuned-Mass Dampersrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 138 no 6 ArticleID 061003 2016

[39] E Lefeuvre D Audigier C Richard and D Guyomar ldquoBuck-boost converter for sensorless power optimization of piezoelec-tric energy harvesterrdquo IEEE Transactions on Power Electronicsvol 22 no 5 pp 2018ndash2025 2007

[40] I Gradshteyn and I Ryzhik Table of Integrals Series andProduct Academic Press 1980

[41] T Asami O Nishihara and A M Baz ldquoAnalytical solutions toh8 and h2 optimization of dynamic vibration absorbers attachedto damped linear systemsrdquo Journal of Vibration amp Acoustics p124 2002

[42] Y Liu L Zuo C-C Lin and J Parker ldquoExact H2 optimaltuning and experimental verification of energy harvesting serieselectromagnetic tuned mass dampersrdquo in Proceedings of Activeand Passive Smart Structures and Integrated Systems 2016March2016

[43] T Haskett B Breukelman J Robinson and J KottelenbergTuned mass dampers under excessive structural excitation2003

[44] AGDavenport ldquoGust loading factorsrdquo Journal of the StructuralDivision vol 93 pp 11ndash34 1967


[45] N Chen Y Li and H Xiang ldquoA new simulation algorithm ofmultivariate short-term stochastic wind velocity field based oninverse fast Fourier transformrdquo Engineering Structures vol 80pp 251ndash259 2014

[46] D C K Poon S Shieh L M Joseph and C Chang ldquoStructuraldesign of taipei 101 the worldrsquos tallest buildingrdquo in Proceedingsof the CTBUH 2004 Seoul Conference 2004


mitigation for TMD-equipped multilayer buildings struc-tures in [11] In addition Ali and Adhikari proposed theidea of an energy harvester dynamic vibration absorber (EH-DVA) [19] in which electric energy is generated from strainsdeveloped in layers of piezoelectric material with the electriccircuit mounted onto the attached vibrating mass of a TMDThey also provided an analytical derivation of the EH-DVAunder harmonic and random excitation [19 21]

Lately an inerter a two-terminal mechanical device isproposed by Smith et al [22 23] The inerter is a devicethat provides a force proportional to the relative accelerationbetween its two terminals Moreover the inerter has beenstudied for improvement of car suspensions [24ndash26] andrailway vehicles [27 28] Other applications of inerters canalso be found in civil engineering for improving the per-formance of vibration control and displacement mitigationin base isolation systems It was proposed the idea of usingan inerter-like ball-screw mechanism to be employed in aviscous mass-damper or tuned viscous mass-damper systemin [29] A type of electromagnetic inertial mass-damperusing a ball-screw mechanism and a motor was presented tocontrol the vibration of structures subjected to earthquakesin [30] Moreover an inertia-based passive control basedon a gyro-mass-damper using gear assemblies was alsostudied as a type of supplemental damper [31] and as abase isolation system [32] Lazar et al [33 34] proposed aninerter-based system with a configuration similar to that ofa TMD which is termed as tuned inerter damper (TID)and developed an analytical tuning rule of a TID based onthe Den Hartogrsquos method Marian and Giaralis [35] alsoproposed the concept of tunedmass-damper-inerter (TMDI)to mitigate the oscillatory motion of support systems whichare stochastically excited They also developed the optimalTMDI parameters for undamped primary structures underwhite noise excitation in closed-form as functions of theTMD mass and the inertance Further they proposed theenergy harvesting-enabled tuned mass-damper-inerter (EH-TMDI) configuration for simultaneous vibration suppressionand energy harvesting [36]

Inspired by the idea of EH-TMDI [35 36] this paperproposes a novel inerter-based dynamic vibration absorbercalled electromagnetic resonant shunt tuned mass-damper-inerter (ERS-TMDI) in which we replace the dissipatedelement of the TMDI with the electromagnetic transducershunted with a resonant R-L-C circuit for simultaneousvibration control and energy harvesting In addition thispaper derives the closed-form solution of optimal designparameters to the ERS-TMDI for building structures underrandom wind and presented numerical analyses 1198672 normoptimal method is employed for the undamped single degreeof freedom (SDOF) structure with the objective to minimizethe mean squared value of the displacement of the primarystructureThe advantages of the ERS-TMDIwill be illustratedthrough the comparison with the classical TMD series-typedouble-mass TMDs [37] and ERS-TMDs [38]

The remainder of this paper is organized as followsIn Section 2 the governing equations of the SDOF systemwith ERS-TMDI are derived Section 3 derives optimal 1198672closed-form solutions to the ERS-TMDI systems under the

wind force excitation In Section 4 the numerical analysis ofstructures is presented in comparison with the classical TMDand ERS-TMDs Conclusions are drawn in Section 5

2 The Single Degree of Freedom SystemModel of the Electromagnetic ResonantShunt Tuned Mass-Damper-Inerter

As is shown in Figure 1(a) for easily understanding thereduction effects of the damper a multistory building issimplified as a general multidegree of freedom (MDOF)system Further Figure 1(a) also shows the configuration ofERS-TMDI for the MDOF where the ERS-TMDI is obtainedby replacing the original energy dissipative damping 119888119879 inclassical TMD with an electromagnetic transducer shuntedwith a resistance 119877 capacitance 119862 and inductance 119871 [16 39]and an inertance b a two-terminal mechanical device isassigned between the absorber and next story of structure forsimultaneous vibration control and energy harvesting underthe force excitation of the primary structure

When multistory buildings are subjected to wind vibra-tion the first-order modal played a very important role inthe analysis of vibration Therefore the primary structure ischaracterized as single degree of freedom (SDOF) structurewith a linear spring of stiffness 119896119904 a mass119898119904 and a dampingcoefficient 119888119904 as shown in Figure 1(b)

The motion equations of the combined SDOF and ERS-TMDI system shown in Figure 1(b) can be given by

119898119904119904 + 119888119904119904 + 119896119904119909119904 minus 119896119879 (119909119879 minus 119909119904) + 119896119891119868 = 119865(119898119879 + 119887) 119879 + 119896119879 (119909119879 minus 119909119904) minus 119896119891119868 = 0119896V ( 119909119879 minus 119909119904) + 119877119868 + 119871 119868 + 1119862 int 119868 119889119905 = 0

(1)

when the primary structure is undamped the above can besimplified in the Laplace domain

1198981199041199091199041199042+119896119904119909119904 minus 119896119879 (119909119879 minus 119909119904) + 119896119891119868 = 119865(119898119879 + 119887) 1199091198791199042 + 119896119879 (119909119879 minus 119909119904) minus 119896119891119868 = 0

119896V (119909119879 minus 119909119904) 119904 + 119877119868 + 119871119868119904 + 119868119862119904 = 0(2)

normalizing the frequency and setting 119904 = 119895120596 we can have

119909119904 (119895120572)2 + 119909119904 minus 1205831198912119879 (119909119879 minus 119909119904) + 119896119891119896119904 119868 =119865119896119904

(1 + 120575) 119909119879 (119895120572)2 + 1198912119879 (119909119879 minus 119909119904) minus 119896119891120583119896119904 119868 = 0119896V (119909119879 minus 119909119904) (119895120572) + 2120577119890119891119890119871119868 + 119871119868 (119895120572) + 1198681198711198912119890 1(119895120572)= 0

(3)


(b)

F

F

F

F

F

kncn

Cnbn

kTn

mTn

In

Ln

Rn

enEMFfn

EMF

mn

knminus1 cnminus1

Cnminus1bnminus1

kTnminus1

Inminus1

Lnminus1

Rnminus1

enminus1EMFfnminus1

EMF

mnminus2

k1c1

C1b1

kT1

m1

I1

L1

R1

e1EMFf1

EMFmT1

kscs

Cb

mT

L

R

eEMFfEMF

ms

I

kT

(a)

mnminus1

mTnminus1



119865119896119904 119909119903 ((1 + 120575) (119895120572)2 + 1198912119879) + 119909119904 (1 + 120575) (119895120572)2 minus 119896119891120583119896119904 119868 = 0119896V119909119903 (119895120572) + 119871119868(2120577119890119891119890 + (119895120572) + 1198912119890 1(119895120572)) = 0

(4)





PIV = 119864 [1199092119904 ]21205871205961199041198780 =

⟨1199092119904⟩21205871205961199041198780 (5)



100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (6)



minusinfin

100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (7)



119883119899 = 119909119904119865119896119904 =1198912119890 1198912119879 + 21198911198901198912119879120577119890 (119895120572) + (1198912119890 120595 + 1198912119879 (1 + 120583119896)) (119895120572)2 + 2119891119890120595120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6

1198600 = 1198912119890 1198912119879 1198601 = 21198912119879119891119890120577119890 1198602 = 1198912119890 120595 + 1198912119879 (1 + 120583119896) + 1198912119890 1198912119879 (1 + 120583120595) 1198603 = 2119891119890120577119890 (120595 + 1198912119879 (1 + 120583120595)) 1198604 = 1198912119879 (1 + 120583119896) (1 + 120583120595) + (1 + 1198912119890 ) 120595 1198605 = 2119891119890120577119890120595 1198606 = 120595(8)




(9)


120597PIV120597119891119879 = 0120597PIV120597120577119890 = 0120597PIV120597119891119890 = 0120597PIV120597120583119896 = 0

(10)



(11a)


(11b)


(11c)


(11d)




119891opt119890 = radic32 + 2512058321205952 + 120583120595 (58 + 119903)2 (16 + 35120583120595 + 1912058321205952)

120577opt119890 = radic(1 (16 + 19120583120595) 120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)))radic16 + 23120583120595 + 812058321205952 (12)


PIoptV


(13)



= 4120583120595 (8 + 10120583120595)16 + 19120583120595


(14)


PIoptlowastV


(15)


119871opt = 119896V119896119891120583opt119896119891opt119879

21198981198791205962119904 119896opt119879 = 119891opt

119879

21198981198791205962119904 119877opt = 2119896V119896119891120577opt119890 119891opt

119890

120583opt119896119891opt119879

2119898119879120596119904

119862opt = 119898119879120583opt119896 119891opt119879

2

119896V119896119891119891opt119890

2

(16)


119891opt = 11 + 120583radic2 + 1205832

120577opt = radic 120583 (4 + 3120583)8 (1 + 120583) (2 + 120583)

(17)



= 32120583 + 40120583216 + 19120583





120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)



119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)


119875 = 1198771198901198682 (21)


119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)


119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)


PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)








08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































(b)

F

F

F

F

F

kncn

Cnbn

kTn

mTn

In

Ln

Rn

enEMFfn

EMF

mn

knminus1 cnminus1

Cnminus1bnminus1

kTnminus1

Inminus1

Lnminus1

Rnminus1

enminus1EMFfnminus1

EMF

mnminus2

k1c1

C1b1

kT1

m1

I1

L1

R1

e1EMFf1

EMFmT1

kscs

Cb

mT

L

R

eEMFfEMF

ms

I

kT

(a)

mnminus1

mTnminus1



119865119896119904 119909119903 ((1 + 120575) (119895120572)2 + 1198912119879) + 119909119904 (1 + 120575) (119895120572)2 minus 119896119891120583119896119904 119868 = 0119896V119909119903 (119895120572) + 119871119868(2120577119890119891119890 + (119895120572) + 1198912119890 1(119895120572)) = 0

(4)





PIV = 119864 [1199092119904 ]21205871205961199041198780 =

⟨1199092119904⟩21205871205961199041198780 (5)



100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (6)



minusinfin

100381610038161003816100381611988311989910038161003816100381610038162 119889120572 (7)



119883119899 = 119909119904119865119896119904 =1198912119890 1198912119879 + 21198911198901198912119879120577119890 (119895120572) + (1198912119890 120595 + 1198912119879 (1 + 120583119896)) (119895120572)2 + 2119891119890120595120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6

1198600 = 1198912119890 1198912119879 1198601 = 21198912119879119891119890120577119890 1198602 = 1198912119890 120595 + 1198912119879 (1 + 120583119896) + 1198912119890 1198912119879 (1 + 120583120595) 1198603 = 2119891119890120577119890 (120595 + 1198912119879 (1 + 120583120595)) 1198604 = 1198912119879 (1 + 120583119896) (1 + 120583120595) + (1 + 1198912119890 ) 120595 1198605 = 2119891119890120577119890120595 1198606 = 120595(8)




(9)


120597PIV120597119891119879 = 0120597PIV120597120577119890 = 0120597PIV120597119891119890 = 0120597PIV120597120583119896 = 0

(10)



(11a)


(11b)


(11c)


(11d)




119891opt119890 = radic32 + 2512058321205952 + 120583120595 (58 + 119903)2 (16 + 35120583120595 + 1912058321205952)

120577opt119890 = radic(1 (16 + 19120583120595) 120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)))radic16 + 23120583120595 + 812058321205952 (12)


PIoptV


(13)



= 4120583120595 (8 + 10120583120595)16 + 19120583120595


(14)


PIoptlowastV


(15)


119871opt = 119896V119896119891120583opt119896119891opt119879

21198981198791205962119904 119896opt119879 = 119891opt

119879

21198981198791205962119904 119877opt = 2119896V119896119891120577opt119890 119891opt

119890

120583opt119896119891opt119879

2119898119879120596119904

119862opt = 119898119879120583opt119896 119891opt119879

2

119896V119896119891119891opt119890

2

(16)


119891opt = 11 + 120583radic2 + 1205832

120577opt = radic 120583 (4 + 3120583)8 (1 + 120583) (2 + 120583)

(17)



= 32120583 + 40120583216 + 19120583





120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)



119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)


119875 = 1198771198901198682 (21)


119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)


119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)


PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)








08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































119883119899 = 119909119904119865119896119904 =1198912119890 1198912119879 + 21198911198901198912119879120577119890 (119895120572) + (1198912119890 120595 + 1198912119879 (1 + 120583119896)) (119895120572)2 + 2119891119890120595120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6

1198600 = 1198912119890 1198912119879 1198601 = 21198912119879119891119890120577119890 1198602 = 1198912119890 120595 + 1198912119879 (1 + 120583119896) + 1198912119890 1198912119879 (1 + 120583120595) 1198603 = 2119891119890120577119890 (120595 + 1198912119879 (1 + 120583120595)) 1198604 = 1198912119879 (1 + 120583119896) (1 + 120583120595) + (1 + 1198912119890 ) 120595 1198605 = 2119891119890120577119890120595 1198606 = 120595(8)




(9)


120597PIV120597119891119879 = 0120597PIV120597120577119890 = 0120597PIV120597119891119890 = 0120597PIV120597120583119896 = 0

(10)



(11a)


(11b)


(11c)


(11d)




119891opt119890 = radic32 + 2512058321205952 + 120583120595 (58 + 119903)2 (16 + 35120583120595 + 1912058321205952)

120577opt119890 = radic(1 (16 + 19120583120595) 120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)))radic16 + 23120583120595 + 812058321205952 (12)


PIoptV


(13)



= 4120583120595 (8 + 10120583120595)16 + 19120583120595


(14)


PIoptlowastV


(15)


119871opt = 119896V119896119891120583opt119896119891opt119879

21198981198791205962119904 119896opt119879 = 119891opt

119879

21198981198791205962119904 119877opt = 2119896V119896119891120577opt119890 119891opt

119890

120583opt119896119891opt119879

2119898119879120596119904

119862opt = 119898119879120583opt119896 119891opt119879

2

119896V119896119891119891opt119890

2

(16)


119891opt = 11 + 120583radic2 + 1205832

120577opt = radic 120583 (4 + 3120583)8 (1 + 120583) (2 + 120583)

(17)



= 32120583 + 40120583216 + 19120583





120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)



119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)


119875 = 1198771198901198682 (21)


119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)


119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)


PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)








08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































119891opt119890 = radic32 + 2512058321205952 + 120583120595 (58 + 119903)2 (16 + 35120583120595 + 1912058321205952)

120577opt119890 = radic(1 (16 + 19120583120595) 120583120595 (7012058331205953 + 24 (4 + 119903) + 4120583120595 (68 + 11119903) + 12058321205952 (247 + 18119903)))radic16 + 23120583120595 + 812058321205952 (12)


PIoptV


(13)



= 4120583120595 (8 + 10120583120595)16 + 19120583120595


(14)


PIoptlowastV


(15)


119871opt = 119896V119896119891120583opt119896119891opt119879

21198981198791205962119904 119896opt119879 = 119891opt

119879

21198981198791205962119904 119877opt = 2119896V119896119891120577opt119890 119891opt

119890

120583opt119896119891opt119879

2119898119879120596119904

119862opt = 119898119879120583opt119896 119891opt119879

2

119896V119896119891119891opt119890

2

(16)


119891opt = 11 + 120583radic2 + 1205832

120577opt = radic 120583 (4 + 3120583)8 (1 + 120583) (2 + 120583)

(17)



= 32120583 + 40120583216 + 19120583





120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)



119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)


119875 = 1198771198901198682 (21)


119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)


119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)


PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)








08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References






















































120583119861opt = 2120583]opt = radic1 + 2120583]119861opt = 11 + 2120583

1205772opt = 01205773opt = 12radic 31205831 + 2120583

(19)



119883119903 = 119909119903119865119896119904 = minus1198912119890 120595 (119895120572)2 + 2120595119891119890120577119890 (119895120572)3 + 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (20)


119875 = 1198771198901198682 (21)


119868119899 = 119868119865119896119904 =(119896V119871) 120595 (119895120572)41198600 + 1198601 (119895120572) + 1198602 (119895120572)2 + 1198603 (119895120572)3 + 1198604 (119895120572)4 + 1198605 (119895120572)5 + 1198606 (119895120572)6 (22)


119875119899 = radic119875119865119896119904 =119868119899radic1198771198901198921205962119904 (23)


PI119890 = 1198962V41198911198901198912119879120583119896120577119890120583 (1 + 120575) 1198712 (24)








08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































08

00 02 04 06 08 10

09

10

11

Opt

imal

fT

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(a)

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution

Opt

imal

휇k

(b)

Opt

imal

fe

0800 02 04 06 08 10

09

10

11

Inertance ratio 훿

휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(c)

Opt

imal

휁 e

00

01

02

03

04

05


휇 = 002 of approx

휇 = 002 of exact

휇 = 01 of approx


solution

solution

solution(d)








0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































0005

01015

02

0

1

205

1

15

2

Mec

hani

cal t

unin

g ra

tiofT


(a)0

00501

01502

0

1

20

05

1

15

coup

ling

coeffi

cien

t휇k

Elec

trom

agne

tic m

echa

nica

l


(b)

0005

01015

02

0

1

208

085

09

095

1

Elec

tric

al d

ampi

ng ra

tiofe


(c)0

00501

01502

0

1

20

02

04

06

08

Elec

tric

al d

ampi

ng ra

tio휁 e


(d)





119878 (120596) = 8120587 11989921199062lowast120596 (1 + 1198992)43 (25a)






119871 117 [H]


119862 102 [F]


119896V 119896119891 150


07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































07 08 09 1 11 12 13123456789

101112

Without TMDERS-TMDs



ERS-TMDI (훿 = 05)

Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80

ERS-TMDs



Nor

mal

ized

disp

lace

men

tXr

(b)


07 08 09 1 11 12 13123456789

101112



Nor

mal

ized

disp

lace

men

tXn

(a)

10

0

20

30

40

50

60

70

80



Nor

mal

ized

disp

lace

men

tXr

(b)


119899 = 60012059612058711988110 (25b)

119906lowast = 119881119911119870ln (1199111199110) (25c)





100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



















































100 200 300 400 500 600 700 800 900 1000

0

5

10

15W

ind

spee

d (m

s)

Time (s)

Wind speed

minus5

minus10

minus15

(a)

001 01 1

001

01

1

10

100

1000

Frequency (Hz)



Win

d sp

ectr

um (m

2s

3)

(b)


0 200 400 600 800 1000 1200

0

20

40

60

Times (s)

Disp

lace

men

t (m

m)


minus20

minus40

minus60

(a)

0 200 400 600 800 1000 1200

0

10

20

30

40

Times (s)

Acce

lera

tion

(mm

s2)

minus10

minus20

minus30

minus40


(b)











0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References





















































0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

Times (s)

P(w

)




5 Conclusions






Acknowledgments


References



























































































Documents

Wind Induced Vibration Control and Energy Harvesting of ... · ResearchArticle Wind Induced Vibration Control and Energy Harvesting of Electromagnetic Resonant Shunt Tuned Mass-Damper-Inerter