High Response Performance of a Tuned-Mass Damper for Vibration

Research ArticleHigh Response Performance of a Tuned-MassDamper for Vibration Suppression of OffshorePlatform under Earthquake Loads

Qiong Wu1 Xilu Zhao1 Rencheng Zheng2 and Keisuke Minagawa1

1College of Mechanical Engineering Saitama Institute of Technology Saitama 369-0293 Japan2Institute of Industrial Science The University of Tokyo Tokyo 153-8505 Japan

Correspondence should be addressed to Qiong Wu wuqiong55126com

Received 14 November 2015 Revised 19 March 2016 Accepted 24 March 2016

Academic Editor Salvatore Strano

Copyright copy 2016 Qiong Wu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Currently tuned-mass dampers (TMDs) are widely applied to maintain the stability of offshore platforms in hostile environmentshowever the stability system of offshore platforms faces considerable challenges under critical earthquake loads of the initialperiod Therefore this study concentrated on the high response performance of a simple passive TMD system and numerical andexperimental investigations were performed using a 1 200-scale prototype The obtained results indicated that the displacementacceleration and their power spectral density all decreased significantly for the offshore platform with the TMD system Byfurther analyses of its high response characteristics it was validated that the TMD reactions can commence within the first 3 sof earthquake excitation while the fundamental natural frequency was consistently tuned for the TMD system dependent on thedynamic magnification factor The evaluation indices also confirmed that this method is effective in reducing the overall vibrationlevel and the maximum peak values of the offshore platform exposed to earthquake excitations mainly because of its high responsecharacteristics

1 Introduction

Offshore platforms of the offshore oil industry are usedfor exploration drilling tender-assisted drilling productionaccommodation and maintenance [1] Such offshore struc-tures which can be located in hostile environments arealways exposed to loadings fromwind waves and sometimesearthquakes the latter being one of the most violent loadingsthese structures might endure Therefore it is necessaryto study the relationship between irregular loadings andthe corresponding response of offshore structures to deriveeffective vibration-suppression systems or techniques [2]

In deep-water environments offshore platforms are sus-ceptible to vibration induced by the action of waves whichnot only affects their structural strength but also has con-siderable impact on their reliability and safety [3] Accidentrates due to the effects of vibration and structural deficienciesremain comparatively high making an accurate analysis ofjack-up behaviors increasingly important Recent attentionhas focused on understanding the behaviors of offshore

platforms under dynamic loading conditions [4] Jack-upplatforms are flexible structures built to have natural periodsof the same order as the predominant wave periods for manyseas which are designed based on structural modeling ofthe legs analysis of the degree of flexibility provided by thespudcan and assessment of the nature of the wave loads [5]These sophisticated structures generally result in self-excitednonlinear hydrodynamic forces and the resulting large defor-mations in turn cause a highly nonlinear response [6]

In recent decades vibration-control technologies imple-mented based on platform type have achieved significantsuccess in mitigating the vibration of land-based structures[7] For example a network-based modeling and activetuned-mass damper mechanism was investigated for an off-shore steel jacket platform which can be significantly capableof reducing the required control force and the oscillationamplitudes of the offshore platform [8] Moreover a typicaltension-leg type of floating platform had been studied whichincorporated a tuned-liquid-column damper [9] RecentlyZhang et al [10] developed a sliding mode 119867

infincontrol to

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7383679 11 pageshttpdxdoiorg10115520167383679

2 Shock and Vibration

improve the control performance of the offshore platformsubject to wave-induced force as well as external disturbanceIn addition compared with the 119867

infincontrol scheme a

network-based modeling and event-triggered 119867infin

reliablecontrol for an offshore structure was proposed [11] It cansuppress the vibration of the offshore structure to almost thesame level as the119867

1controller while the former requires less

control cost Specifically a more general uncertain dynamicmodel of the offshore platform was developed and thena novel delayed sliding mode control scheme using mixedcurrent and delayed states was proposed in [12] It is shownthrough the simulating results that this scheme is moreeffective in both improving the control performance andreducing control force of the offshore platform

Gattulli and Ghanem [13] developed an active massdamper control technique for the suppression of vortex-induced vibrations in offshore structures and Kawano [14]investigated semiactive control devices applied in jack-upoffshore platforms in which the active control force isdetermined using a time-domain transient optimal controlmethod Zhou et al [15] proposed a semiactive controlmethod utilizing the energy-dissipation principle and a bang-bang control based on a linear quadratic regulator optimalcontrol theory Amongst the various semiactive controldevices a magnetorheological damper has been studiedfor structural vibration control and the simulated resultsindicated that such a system could provide quick reactionwith little time delay insensitivity to temperature and smallpower requirement [16]

However offshore platforms are located in ocean envi-ronments which could increase the cost of maintenance ofactive and semiactive systems and indirectly increase the riskto staffTherefore passive vibration controlmight be themostsuitable and feasible strategy for vibration control in offshoreplatforms

A passive TMDwas studied for the reduction of vibrationin flexible structures subjected to long-reduction narrowexcitations [17] This was conducted because amongst thenumerous passive control techniques the TMD is one ofthe simplest control devices [18] The determination of theoptimal parameters has been performed according to thedifferent objectives of reducing the maximum displacementsstory drifts and base shear for different harmonic whitenoise and earthquake excitations [19] Li and Zhu [20]proposed using double TMDs in which the mass damperwas optimized for high effectiveness and robustness inreducing undesirable vibrations under the effects of groundacceleration Bekdas and Nigdeli [21] found a mathematicaloptimization method using harmony search which wasapplied successfully to passive TMDs Yu et al [22] outlineda robust design optimization framework dedicated to TMDs

Many researchers have studied the applicability of TMDsto ground-based structures subject to seismic excitation [23ndash25] For an earthquake excitation in which its duration issubstantially shorter considerable disasters often occurredduring the initial period of an earthquake load The sev-eral previous studies used the tuned-mass damper withviscous damping to improve the vibration-control effective-ness However because the high response performance oftuned-mass damper is not considered these dampers maynot have enough reaction time to produce a significant

TMD

Offshore platform

Figure 1 Schematic of a jack-up offshore platform with a tuned-mass damper (TMD)

effect in the initial seconds of the earthquake excitationTsai [26] simulated TMDs with different parameters underdifferent earthquake excitations It was found that a TMDwith damping had little reaction to the structural response inthe initial seconds following the occurrence of an earthquaketherefore a virtual accelerator was proposed to improve theTMD performance Lin et al [27] studied an undampedTMD with a resettable variable stiffness device which couldavoid detuning effects and assure optimal control perfor-mance Zhang and Balendra [28] investigated the feasibilityof adopting a TMD for the control of an inelastic structuresubject to seismic motions Marano et al [29] dealt with theoptimal design of a TMD to reduce undesirable vibrationaleffects which were originated in linear structures by seismicexcitations However based on our literature review passiveTMD technology has had little application to the control ofthe response of offshore platforms to an earthquake becauseof the complexity of the excitation Many control strategieshave been shown effective in the mitigation of structuralvibration however few studies have examined the responseperformance of a TMD at the onset of earthquake excitationIn order to reduce the risk of damage to jack-up offshoreplatforms in harshmarine environments studies are requiredto develop efficient and practical vibration-control strategiesthat can suppress the dynamic response of offshore structures

This study involved comprehensive experimental andanalytical investigations to extend the understanding of thehigh response performance of TMD systems under twoseismic stimuliThe experimental processwas based on a pro-totype in an experimental setting In the following sectionsthe results of the amplitude and frequency responses relativemotion high response characteristics and evaluation indicesare interpreted In addition the tuned-mass damper controlschemewill be comparedwith some existing control schemessuch as tuned-liquid damper [30] and active mass dampercontrol scheme [31]

2 Methods and Materials

21 Modeling To analyze the effectiveness of a TMD anactual jack-up offshore platform (Bohai number 5 located inthe Southern Sea) [32] was considered as the research targetAs demonstrated in Figure 1 the jack-up offshore platform

Shock and Vibration 3

m1

m2

k1k1

k2k2

c1

xV

Figure 2 Systemic modeling of a jack-up offshore platform with atuned-mass damper (TMD)

comprises a rectangular platform resting on four independentoperating legs with a TMD attached beneath the platform

TheTMDdevice consists of a frame amass two springs fourwheels and two tracks The size of the working platform is575 times 340 times 550m the length and diameter of the operatinglegs are 78 and 3m respectively and the limit of the operatingwater depth is 40m

As presented in Figure 2 a systemic model of a jack-up offshore platform with a TMD can be considered as ageneral structure (119898

1+ 1198982) that includes the main structure

of the offshore platform (1198981) and the substructure of the

TMD (1198982) Thereby the dynamic model can be expressed as

11989811+ 11988811+ 11989611199091minus 1198962(1199092minus 1199091) = minus119898

1119881

11989822+ 1198962(1199092minus 1199091) = minus119898

2119881

(1)

where 1198981 1198881 and 119896

1are the mass damping and stiffness

of the main structure respectively 1198982and 119896

2are the mass

and stiffness of the substructure respectively and 119881is the

acceleration vector of the seismic loadsTo resolve (1) while 119909

1= 1198831119890119894119908119905 and 119909

2= 1198832119890119894119908119905

[33] the imaginary amplitudes of the main structure 1198831and

substructure1198832can be expressed as

1198831=100381610038161003816100381610038161198831119890119894119908119905

10038161003816100381610038161003816=

1198981119881(1198962minus 11989821199082

)

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(2)

1198832=100381610038161003816100381610038161198832119890119894119908119905

10038161003816100381610038161003816=

11989821198811198962

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(3)

where 119908 is the excitation frequency of the seismic loadsFrom (2) the dynamic magnification factor for the main

structure can be described as10038161003816100381610038161003816100381610038161003816

1198831

119883st

10038161003816100381610038161003816100381610038161003816

= radic(1205782

minus 1205822

)2

[(1205782 minus 1205822) (1 minus 1205822) minus 12058312057821205822]2

+ [21205851120582 (1205782 minus 1205822)]

2

(4)

where 119883st = 11989811198811198961is the static displacement of the main

structure 120578 = 11990821199081is the natural frequency ratio 119908

1=

radic11989611198981and 119908

2= radic11989621198982are the natural frequencies of the

main structure and substructure respectively 120582 = 1199081199081is

the ratio of the excitation and natural frequencies of the mainstructure and 120585

1= 1198881211989811199081is the damping ratio of themain

structureBased on (4) the dynamic magnification factor becomes

zero while 120578 = 120582 which indicates that optimal vibrationreduction will be achieved when exposed to the specificexcitation frequencies of the seismic loads The seismic loadsinvolve different frequency components which can resultin the maximum amplitude vibration of the main structure

during the range of the resonance frequencyTherefore whenthe natural frequency of the TMD is adjusted to the naturalfrequency of the structure the TMD will be in a resonantstate Therefore a large amount of the structural vibrationenergy will be transferred to the TMD and thus vibrationreduction can be achieved Bymodeling the offshore platformwith a fixed TMD the natural frequency can be measuredduring experimental testing

22 Numerical Analysis By application of the central differ-encemethod to solve (1) [34] using time step 119894 the differentialacceleration and velocity can be expressed as

(119894)

=119909(119894+1)

minus 2119909(119894)

+ 119909(119894minus1)

Δ2119905

(119894)

=119909(119894+1)

minus 119909(119894minus1)

2Δ119905

(5)

where Δ119905 is the time increment When (5) are substitutedinto (1) the formula for the calculation of the displacementdifference can be expressed as


1199091

(119894+1)

=

(21198981minus 1198961Δ2

119905 minus 1198962Δ2

119905) 1199091

(119894)

+ 1198962Δ2

1199051199092

(119894)

+ (minus1198981+ 05119888

1Δ119905) 1199091

(119894minus1)

minus 1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(119894+1)

=

(21198982minus 1198962Δ2

119905) 1199092

(119894)

+ 1198962Δ2

1199051199091

(119894)

minus 11989821199092

(119894minus1)

minus 1198982119881Δ2

119905

1198982

(6)

The initial conditions of the offshore platform can beexpressed as

1199091

(0)

= 0

1

(0)

= 0

1

(0)

= 0

1199092

(0)

= 0

2

(0)

= 0

2

(0)

= 0

(7)

When the initial conditions of (7) are substituted into (5)the two initial displacements will be zero for example

1199091

(1)

= 0

1199092

(1)

= 0

(8)

Inserting the initial displacements of the second step1199091

(0)

= 0 1199092

(0)

= 0 1199091

(1)

= 0 and 1199092

(1)

= 0 into (6) meansthat the subsequent differential displacements 119909

1

(2) and 1199092

(2)

can be obtained By continually increasing the calculationstep the differential displacements of 119909

1

(119894+1) and 1199092

(119894+1) canbe estimated from 119909

1

(119894minus1) 1199092

(119894minus1) 1199091

(119894) and 1199092

(119894)

23 Evaluation Index A root mean square (RMS) of thedisplacement or acceleration response can be expressed as

RMS = radic1

119899

119899

sum

119894=1

1199102

119894 (9)

where 119910119894is the sampling value of the displacement or

acceleration responseTo evaluate effectiveness of the vibration reduction dur-

ing the entire earthquake period an evaluation index wasproposed

119869 =RMSctrlRMSunctrl

(10)

where RMSctrl and RMSunctrl are the RMS values of thedisplacement or acceleration responses of the main structurefor the entire earthquake period for the cases with andwithout the TMD system respectively

To evaluate the maximum response reduction a maxi-mum peak value was selected as 119910max = max119910

119894 Then a

relative ratio of the maximum peak values can be obtainedfrom

120573 =119910max-unctrl minus 119910max-ctrl

119910max-unctrl (11)

where 119910max-ctrl and 119910max-unctrl are the maximum peak valuesof the displacement or acceleration responses of the mainstructure with and without the TMD system respectively

3 Experiment

31 Apparatus As shown in Figure 3 the testing systemcomprised a personal computer vibration signal generatoramplifier shaker offshore platform system acceleration sen-sor laser displacement sensor and fast Fourier transformanalyzer In the experiment the sand height was 80mm andthe water depth was 400mm The mass of the offshore plat-form as the main structure (119898

1) was 2346 kg and the mass of

the TMD was 0591 kg Through experimental validation thedamping coefficient was determined as 0012

As shown in Figure 4 a 1 200-scale four-column type ofoffshore platform was constructed The operating platformwas simplified to a horizontal rectangular metal mass Thesupporting columns were simplified to hollow tubes Cylin-drical pile shoes were set at the roots of the columns and theconnections between the columns and the operating platformwere rigidTheoffshore platformwas placed in a tank thatwasfixed on the shaker which simulated the seismic loads

32 Process Initially the substructure of the TMD wasinstalled at the bottomof the offshore platform tomeasure thefirst natural frequency of the complete structure The inputsignal was a 0ndash8Hz sweep signal

The second step was to remove the substructure from theoffshore platform and to attach it to the shaker Then thespring stiffness of the TMD was adjusted to make the firstnatural frequency of the substructure the same as that of themain structure As shown in Figure 5 the peak frequencyresponse function of both was 256Hz

In the third step two types of seismic wave (El-CentroNS and Taft EW) [35] were generated by the signal generatorand fed to the shaker to evaluate the effectiveness of thevibration-control system The recording time was 25 s andthe sampling frequency was 50Hz for the two seismic wavesConsequently the displacement and acceleration of the mainstructure and the relative displacement between the mainstructure and the substructure were recorded


TMD

ShakerAmplifierSignal generator

FFT analyzer Displacement sensor

Water

Sand

PC

Acceleration sensor

Figure 3 Diagram of testing system (PC personal computerTMD tuned-mass damper and FFT analyzer fast Fourier transformanalyzer)

Platform

TMD

Shaker

Tank

(a) Offshore platform system

Spring

Track Wheel Frame

Mass

(b) Tuned-mass damper

Figure 4 Experimental setting (a) offshore platform and (b) tuned-mass damper (TMD)

Here the El-Centro NS was the NS component recordedat the Imperial Valley Irrigation District substation in ElCentro California USA on May 18 1940 Its magnitude was69 and the peak acceleration was 341 cms2 The Taft EWwas the EW component recorded at Kern County CaliforniaUSA on July 21 1952 Its magnitude was 77 and the peakacceleration value was 1759 cms2 [36]

0

2

4

6

8

10

12

0 1 2 3 4

TMDPlatform

Frequency (Hz)

Freq

uenc

y re

spon

se fu

nctio

n (d

B)

Figure 5 Frequency response analysis of the tuned-mass damper(TMD) and offshore platform

4 Results

41 Amplitude Response Analysis Figures 6 and 7 display thetime series of the responses of the main structure with andwithout the TMD under the excitation of the El-Centro NSand Taft EW seismic waves The experimental results of thedisplacement response are shown in Figures 6(a) and 6(c)and the numerical values of the displacement response areshown in Figures 6(b) and 6(d) The experimental results ofthe acceleration response are shown in Figure 7

The numerical analyses show that the peak values of thedisplacement responses decreased significantly with TMDcontrol compared with those cases without TMD control Itcan be seen that this decreasing tendency was more signifi-cant for the acceleration responses The results of the exper-imental analyses are consistent with the numerical analysesand the accuracy of the numerical method is verified by thesimilarity between the amplitudes obtained from the simu-lation and the experiment However the numbers of wavepeaks obtained in the numerical analyses are greater than inthe experimental analyses because of the ideal conditionalsetting of the numerical method These results indicate thatthe control performance of the TMD is as effective as anenergy-dissipation device for the reduction of the mainstructural response

It is particularly important for vibration suppressionunder excitation by seismic waves that the TMD can effec-tively reduce the relatively high-amplitude displacements andaccelerations These high-amplitude displacements occurredduring the early period of the test and the TMD respondedquickly to these early vibration excitations For the relativelylow-amplitude displacements and accelerations the TMDcontrol resulted in little effective reduction however theselow-amplitude locals provided only a small contribution tothe platform vibration

42 Frequency Response Analysis Based on a frequencyresponse analysis the power spectral density (PSD) of thedisplacement of the main structure is presented in Figure 8and the PSD of the acceleration of the main structure is


minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)

No TMD controlTMD control

(a) Numerical analysis under the El-Centro NS seismic wave (TMDtuned-mass damper)

minus30minus20minus10

0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)


(b) Experimental result under the El-Centro NS seismic wave (TMDtuned-mass damper)


0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)


(c) Numerical analysis under the Taft EW seismic wave (TMD tuned-mass damper)

0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



(d) Experimental result under the Taft EW seismic wave (TMD tuned-mass damper)

Figure 6 Displacement responses under the El-Centro NS and Taft EW seismic waves (a) numerical analysis under the El-Centro NS (b)experimental result under the El-Centro NS (c) numerical analysis under the Taft EW and (d) experimental result under the Taft EW

minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)

(a) Experimental result under the El-Centro NS seismic wave (TMDtuned-mass damper)

135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)

(b) Experimental result under the Taft EW seismic wave (TMD tuned-mass damper)

Figure 7 Experimental results of the acceleration responses under (a) the El-Centro NS and (b) the Taft EW seismic waves

presented in Figure 9 It is obvious that the high amplitudeof the PSD is around 256Hz for the case without TMDcontrol because the resonance reaction of the main structureoccurred around this frequency domain this can explainwhy the designed TMD frequency was nearly 256Hz asmentioned in relation to Figure 5 Therefore in the casewith TMD control it is significantly effective in reducing thevibration response of the main structure in particular thepeak responses are reduced considerably around the 256Hzfrequency domain

It is validated that the overall vibration response can bedecreased significantly by reducing the first-mode vibrationand these results of the PSD curves explain why the timeseries response is effective in vibration reduction Conse-quently a single damper tuned to the fundamental mode

is adequate for reducing the structural vibration underearthquake excitations Investigation of frequency regionsother than the fundamental frequency revealed no negativeeffects in the nondominant frequency regions

43 Evaluation Index Thecontrol performancewas analyzedby application of the indices of 120573 and 119869 which are definedin (10) and (11) respectively The 120573 index is the ratio ofthe RMS values of the displacements or accelerations of theoffshore platform between the cases with and without theTMD control The 119869 index is the relative ratio of the peakvalues between the cases with and without the TMD control

As shown in Table 1 the 119869 values for the displacementresponse are gt080 which indicates that the vibration of


0

4

8

12

16

0 1 2 3 4Frequency (Hz)


Pow

er sp

ectr

al d

ensit

y (m

2middots)

(a) Power spectral density under the El-Centro NS seismic wave (TMDtuned-mass damper)

0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)

(b) Power spectral density of the Taft EW seismic wave (TMD tuned-mass damper)

Figure 8 Power spectral density of the displacement under (a) theEl-Centro NS and (b) the Taft EW seismic waves

Table 1 Results of the evaluation indices

Seismic excitation Displacement response Acceleration response119869 120573 119869 120573

El-Centro NS 08049 02805 07722 03239Taft EW 08191 02744 07283 04406

the platform was improved significantly during the entireearthquake period when the TMD was used Moreoveranalysis of the 120573 values shows that a reduction of gt27was accomplished for the peak displacement response by theapplication of the TMD system

For the acceleration displacement response the 119869 valuesare gt070 which means that the dynamic performance wasalso improved considerably throughout the entire earthquakeperiod Similarly the 120573 values are gt030 which means thepeak response was decreased by 30 by the application of theTMD system

5 Discussion

It can be seen from the above experimental results that theTMD system can effectively suppress the seismic motionfor an offshore platform model In order to further theunderstanding of the dynamic performances of the TMD

0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)

(a) Power spectral density of the acceleration under El-Centro NSseismic wave (TMD tuned-mass damper)

0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)

(b) Power spectral density of the acceleration under Taft EW seismicwave (TMD tuned-mass damper)

Figure 9 Power spectral density of the acceleration under (a) theEl-Centro NS and (b) the Taft EW seismic waves

a numerical simulation and experimental investigation wereprovided about high response characteristics of the TMD andrelative motion between the offshore platform and TMD

51 High Response Characteristics The duration of an earth-quake excitation is generally short and the maximum influ-ence on a platformrsquos deformation mainly results from itsinitial seconds Therefore it is critical that the high responsespeed of the TMD occurs within the initial seconds of theexcitation To overcome this problem this study proposeda passive TMD without damper for improving the highresponse performance under critical earthquake loads

By application of the central difference method thedisplacements of the substructure and the main structureduring the first 3 s of an excitation can be obtainedThe initialconditions of main structure can be expressed as

1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform

(a) Displacement responses under El-Centro NS seismic wave

minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform

(b) Displacement responses under Taft EW seismic wave

Figure 10 Displacement responses of the tuned-mass damper(TMD) and platform under (a) the El-Centro NS and (b) the TaftEW seismic waves

When the initial conditions of (12) are substituted into(6) the next-step differential displacements 119909

1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)

The ratio of 1199091

(2) and 1199092

(2) can be expressed as

1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)

It can be observed from (14) at the initial time that thedisplacement of the substructure 119909

2

(2) is always greater thanthe displacement of the main structure 119909

1

(2) that is 1199092

(2)

gt

1199091

(2) Based on this theoretical analysis an experiment wasundertaken to investigate this high response phenomenon

As illustrated in Figures 10 and 11 when the mainstructure begins to move under the earthquake excitationthe substructure also moves immediately In addition the

minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)

(a) Acceleration responses under El-Centro NS seismic wave

0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)

(b) Acceleration responses under Taft EW seismic wave

Figure 11 Acceleration responses of the tuned-mass damper (TMD)and platform under (a) the El-Centro NS and (b) the Taft EWseismic waves

displacement and acceleration of the substructure are bothconsiderably greater than themain structureThis reveals thatthe substructure can achieve vibration suppression throughits high response to the seismic excitations

To understand the high response characteristics betteran FFT for the displacement responses was performed andthen the phase differences between the main structure andthe substructure were calculatedThe phase differences underthe two seismic vibrations are presented in Figure 12 It canbe observed that initial phase differences exist in differentfrequency domains although they are only significantly highat frequencies other than around 1-2Hz

52 Relative Motion In order to understand the dynamicperformance more fully it is necessary to study the relativemotion between themain structure and the substructureThemodel of the relative motion is shown in Figure 13 where1199091and 119909

2are the displacements of the main structure and

the substructure respectively The motion between the mainstructure and the substructure can be divided into two partsno relative motion (|119909

2minus 1199091| = 0) and relative motion

(|1199092minus 1199091| = 0) as illustrated in Figures 13(a) and 13(b)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)

(a) Phase difference under El-Centro NS seismic wave

minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)

(b) Phase difference under Taft EW seismic wave

Figure 12 Phase differences of the tuned-mass damper (TMD) andplatform under (a) the El-Centro NS and (b) the Taft EW seismicwaves

respectively The elastic potential energy of the TMD can bewritten as

119864119896=1

21198962(1199092minus 1199091)2

(15)

When |1199092minus 1199091| = 0 the elastic potential energy becomes

zero which indicates that the total seismic vibrational energyis absorbed by the motion of the main structure When |119909

2minus

1199091| = 0 part of the seismic vibrational energy is transferred

to the elastic potential energy of the substructure In thiscase the relative motion can dissipate the partial seismicvibrational energy

The relative displacements under the two seismic exci-tations are presented in Figure 14 It can be observed thatintense relative motion exists throughout most of the timeseries This type of intense relative motion produces elasticdeformation of the spring which can transfer part of theseismic energy to elastic potential energy

53 Effectiveness Comparison Up to now there were fewstudies on the earthquake controlling for the jacket offshoreplatform in particular there are few researches concerningthe high response performance of TMD system Thereforeonly two controllers of tuned-liquid damper [30] and anactive mass damper [31] as the existing control schemeswere found for effectiveness comparison of the tuned-massdamper

In Table 2 the reduction percentages are listed to comparetheir effectiveness about the peak oscillation amplitudes ofthe offshore platform It can be observed that the reductionpercentages of the tuned-mass damper applied in this studyare relatively higher than that of the tuned-liquid damperand active mass damper In fact it is difficult to compare

x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0

Figure 13 Diagrams showing different relative motions (a) norelativemotion (|119909

2minus1199091| = 0) and (b) relativemotion (|119909

2minus1199091| = 0)

Table 2 Effectiveness comparison of the control schemes

Earthquake Controllers Reduction ()Displacement Acceleration

El CentroTuned-liquid damper 176 1128Active mass damper 167 minus677Tuned-mass damper 2805 3239

them under a relatively fair condition due to their differentexperimental conditions However it is indicated that thehigh response characteristics can improve the earthquakecontrolling for the TMD control scheme in this study

6 Conclusions

This study focused on vibration suppression of an offshoreplatform under earthquake loads by application of a TMDThe principles of the relevantmathematical modeling tuningprinciple of the dynamic magnification factor central differ-ence method for numerical analysis and evaluation indiceswere described and an experimental system was constructedbased on a 1 200-scale model of an actual four-column jack-up offshore platform Specific experiments were performedfor the offshore platform both with and without the TMDsystem under two types of seismic load The effectivenessof the TMD system was investigated comprehensively by


minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)

(a) Relative displacement under the El-Centro NS seismic wave

0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)

(b) Relative displacement under the Taft EW seismic wave

Figure 14 Relative displacements under (a) the El-Centro NS and(b) the Taft EW seismic waves

analyses of the amplitude and frequency response relativemotion high response characteristics and evaluation indices

Analyses of the amplitude and frequency responses indi-cated that the displacement acceleration and their powerspectral density decreased significantly for the offshore plat-form with the TMD system Further analyses of the highresponse and relative motion characteristics showed that thedamping reaction started within the first 3 s of earthquakeexcitation even only the fundamental nature frequency wasconsistently tuned for the TMD system Consequently theRMS ratios reached around 80 for the displacements andaround 70 for the acceleration responses The vibratingexcitations of the offshore platform were improved sig-nificantly throughout the entire earthquake period Therelative ratio of the maximum peak values verified that areduction of gt27 was achieved for the maximum peakdisplacement response and a reduction of gt32was achievedfor the acceleration response Based on the numerical andexperimental investigations of this study it is verified thata passive TMD constitutes a simple but feasible measurefor vibration suppression of offshore platforms exposed toearthquake excitations mainly because of its high responsecharacteristics

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thework described in this paper was supported by theGrant-in-Aid for Scientific Research (c) (Grant no 24560271)

References

[1] J Vazquez R Michel J Alford M Quah and K S FooJack Up Units a Technical Primer for the Offshore IndustryProfessional Bennett amp Associates Offshore Technology Devel-opment 2005

[2] K T Chen C H Chou S H Chang and Y H Liu ldquoIntelligentactive vibration control in an isolation platformrdquo AppliedAcoustics vol 69 no 11 pp 1063ndash1084 2008

[3] H Yu X Li and S Yang ldquoDynamic analysis method of offshorejack-up platforms in regular and random wavesrdquo Journal ofMarine Science and Application vol 11 no 1 pp 111ndash118 2012

[4] R J Hunt and P DMarsh ldquoOpportunities to improve the oper-ational and technical management of jack-up deploymentsrdquoMarine Structures vol 17 no 3-4 pp 261ndash273 2004

[5] M S Williams R S G Thompson and G T Houlsby ldquoNon-linear dynamic analysis of offshore jack-up unitsrdquo Computersand Structures vol 69 no 2 pp 171ndash180 1998

[6] M J Terro M S Mahmoud and M A Rohman ldquoMulti-loopfeedback control of offshore steel jacket platformsrdquo Computersand Structures vol 70 no 2 pp 185ndash202 1999

[7] G W Housner L A Bergman T K Caughey et al ldquoStruc-tural control past present and futurerdquo Journal of EngineeringMechanics vol 123 no 9 pp 897ndash971 1997

[8] B Zhang and Q Han ldquoNetwork-based modelling and activecontrol for offshore steel jacket platform with TMD mecha-nismsrdquo Journal of Sound and Vibration vol 333 no 25 pp6796ndash6814 2014

[9] H H Lee S Wong and R Lee ldquoResponse mitigation on theoffshore floating platform system with tuned liquid columndamperrdquoOcean Engineering vol 33 no 8-9 pp 1118ndash1142 2006

[10] B Zhang L Ma and Q Han ldquoSliding mode Hinfin

control foroffshore steel jacket platforms subject to nonlinear self-excitedwave force and external disturbancerdquo Nonlinear Analysis RealWorld Applications vol 14 no 1 pp 163ndash178 2013

[11] B Zhang Q Han and X Zhang ldquoEvent-triggered119867infinreliable

control for offshore structures in network environmentsrdquo Jour-nal of Sound and Vibration vol 368 no 25 pp 1ndash21 2016

[12] B Zhang Q Han X Zhang and X Yu ldquoSliding mode controlwith mixed current and delayed states for offshore steel jacketplatformsrdquo IEEE Transactions on Control Systems Technologyvol 22 no 5 pp 1769ndash1783 2014

[13] V Gattulli and R Ghanem ldquoAdaptive control of flow-inducedoscillations including vortex effectsrdquo International Journal ofNon-Linear Mechanics vol 34 no 5 pp 853ndash868 1999

[14] K Kawano ldquoActive control effects on dynamic response ofoffshore structuresrdquo in Proceedings of the 3rd InternationalOffshore and Polar Engineering Conference pp 594ndash598 June1993

[15] Y Zhou L Xu and Z Li ldquoSeismic response of semi-activecontrol using magneto-rheological fluid dampersrdquo Journal ofCivil Engineering vol 34 no 5 pp 10ndash14 2001

[16] J Ou X Long Q S Li and Y Q Xiao ldquoVibration control ofsteel jacket offshore platform structures with damping isolationsystemsrdquo Engineering Structures vol 29 no 7 pp 1525ndash15382007

[17] J L Almazan J C De la Llera J A Inaudi D Lopez-Garcıaand L E Izquierdo ldquoA bidirectional and homogeneous tunedmass damper a new device for passive control of vibrationsrdquoEngineering Structures vol 29 no 7 pp 1548ndash1560 2007


[18] S Chakraborty and B K Roy ldquoReliability based optimumdesign of Tuned Mass Damper in seismic vibration control ofstructures with bounded uncertain parametersrdquo ProbabilisticEngineering Mechanics vol 26 no 2 pp 215ndash221 2011

[19] M Domizio D Ambrosini and O Curadelli ldquoPerformance oftuned mass damper against structural collapse due to near faultearthquakesrdquo Journal of Sound and Vibration vol 336 pp 32ndash45 2015

[20] C Li and B Zhu ldquoEstimating double tuned mass dampers forstructures under ground acceleration using a novel optimumcriterionrdquo Journal of Sound and Vibration vol 298 no 1-2 pp280ndash297 2006

[21] G Bekdas and S M Nigdeli ldquoEstimating optimum parametersof tuned mass dampers using harmony searchrdquo EngineeringStructures vol 33 no 9 pp 2716ndash2723 2011

[22] H Yu F Gillot and M Ichchou ldquoReliability based robustdesign optimization for tunedmass damper in passive vibrationcontrol of deterministicuncertain structuresrdquo Journal of Soundand Vibration vol 332 no 9 pp 2222ndash2238 2013

[23] T Pinkaew P Lukkunaprasit and P Chatupote ldquoSeismiceffectiveness of tuned mass dampers for damage reduction ofstructuresrdquo Engineering Structures vol 25 no 1 pp 39ndash462003

[24] A Ghosh and B Basu ldquoEffect of soil interaction on theperformance of tuned mass dampers for seismic applicationsrdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1079ndash10902004

[25] C Lin J Ueng and T Huang ldquoSeismic response reductionof irregular buildings using passive tuned mass dampersrdquoEngineering Structures vol 22 no 5 pp 513ndash524 2000

[26] H Tsai ldquoThe effect of tuned-mass dampers on the seismicresponse of base-isolated structuresrdquo International Journal ofSolids and Structures vol 32 no 8-9 pp 1195ndash1210 1995

[27] C Lin L Lu G Lin and T Yang ldquoVibration control ofseismic structures using semi-active friction multiple tunedmass dampersrdquo Engineering Structures vol 32 no 10 pp 3404ndash3417 2010

[28] Z Zhang and T Balendra ldquoPassive control of bilinear hystereticstructures by tuned mass damper for narrow band seismicmotionsrdquo Engineering Structures vol 54 pp 103ndash111 2013

[29] G C Marano R Greco F Trentadue and B Chiaia ldquoCon-strained reliability-based optimization of linear tuned massdampers for seismic controlrdquo International Journal of Solids andStructures vol 44 no 22-23 pp 7370ndash7388 2007

[30] Q Jin X Li N Sun J Zhou and J Guan ldquoExperimental andnumerical study on tuned liquid dampers for controlling earth-quake response of jacket offshore platformrdquoMarine Structuresvol 20 no 4 pp 238ndash254 2007

[31] J POu X LongQ S Li andYQ Xiao ldquoExperimental researchon AMD control of structural vibration of offshore platformrdquoHigh Technology Letters no 10 pp 85ndash90 2002 (Chinese)

[32] D Xie Introduction to Offshore Platform Design HuazhongUniversity of Technology 2008

[33] K Seto Dynamic Vibration Absorber and Its ApplicationCORONA Publishing 2010

[34] S S RaoMechanical Vibration Prentice Hall 5th edition 2010[35] National Research Institute for Earth Science and Disaster Pre-

vention January 2015 httpwwwkyoshinbosaigojpkyoshinquake

[36] I Takewaki A Moustafa and K Fujita Improving the Earth-quake Resilience of BuildingsTheWorst CaseApproach SpringerSeries in Reliability Engineering 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



improve the control performance of the offshore platformsubject to wave-induced force as well as external disturbanceIn addition compared with the 119867

infincontrol scheme a

network-based modeling and event-triggered 119867infin

reliablecontrol for an offshore structure was proposed [11] It cansuppress the vibration of the offshore structure to almost thesame level as the119867

1controller while the former requires less

control cost Specifically a more general uncertain dynamicmodel of the offshore platform was developed and thena novel delayed sliding mode control scheme using mixedcurrent and delayed states was proposed in [12] It is shownthrough the simulating results that this scheme is moreeffective in both improving the control performance andreducing control force of the offshore platform

Gattulli and Ghanem [13] developed an active massdamper control technique for the suppression of vortex-induced vibrations in offshore structures and Kawano [14]investigated semiactive control devices applied in jack-upoffshore platforms in which the active control force isdetermined using a time-domain transient optimal controlmethod Zhou et al [15] proposed a semiactive controlmethod utilizing the energy-dissipation principle and a bang-bang control based on a linear quadratic regulator optimalcontrol theory Amongst the various semiactive controldevices a magnetorheological damper has been studiedfor structural vibration control and the simulated resultsindicated that such a system could provide quick reactionwith little time delay insensitivity to temperature and smallpower requirement [16]

However offshore platforms are located in ocean envi-ronments which could increase the cost of maintenance ofactive and semiactive systems and indirectly increase the riskto staffTherefore passive vibration controlmight be themostsuitable and feasible strategy for vibration control in offshoreplatforms

A passive TMDwas studied for the reduction of vibrationin flexible structures subjected to long-reduction narrowexcitations [17] This was conducted because amongst thenumerous passive control techniques the TMD is one ofthe simplest control devices [18] The determination of theoptimal parameters has been performed according to thedifferent objectives of reducing the maximum displacementsstory drifts and base shear for different harmonic whitenoise and earthquake excitations [19] Li and Zhu [20]proposed using double TMDs in which the mass damperwas optimized for high effectiveness and robustness inreducing undesirable vibrations under the effects of groundacceleration Bekdas and Nigdeli [21] found a mathematicaloptimization method using harmony search which wasapplied successfully to passive TMDs Yu et al [22] outlineda robust design optimization framework dedicated to TMDs

Many researchers have studied the applicability of TMDsto ground-based structures subject to seismic excitation [23ndash25] For an earthquake excitation in which its duration issubstantially shorter considerable disasters often occurredduring the initial period of an earthquake load The sev-eral previous studies used the tuned-mass damper withviscous damping to improve the vibration-control effective-ness However because the high response performance oftuned-mass damper is not considered these dampers maynot have enough reaction time to produce a significant

TMD

Offshore platform

Figure 1 Schematic of a jack-up offshore platform with a tuned-mass damper (TMD)

effect in the initial seconds of the earthquake excitationTsai [26] simulated TMDs with different parameters underdifferent earthquake excitations It was found that a TMDwith damping had little reaction to the structural response inthe initial seconds following the occurrence of an earthquaketherefore a virtual accelerator was proposed to improve theTMD performance Lin et al [27] studied an undampedTMD with a resettable variable stiffness device which couldavoid detuning effects and assure optimal control perfor-mance Zhang and Balendra [28] investigated the feasibilityof adopting a TMD for the control of an inelastic structuresubject to seismic motions Marano et al [29] dealt with theoptimal design of a TMD to reduce undesirable vibrationaleffects which were originated in linear structures by seismicexcitations However based on our literature review passiveTMD technology has had little application to the control ofthe response of offshore platforms to an earthquake becauseof the complexity of the excitation Many control strategieshave been shown effective in the mitigation of structuralvibration however few studies have examined the responseperformance of a TMD at the onset of earthquake excitationIn order to reduce the risk of damage to jack-up offshoreplatforms in harshmarine environments studies are requiredto develop efficient and practical vibration-control strategiesthat can suppress the dynamic response of offshore structures

This study involved comprehensive experimental andanalytical investigations to extend the understanding of thehigh response performance of TMD systems under twoseismic stimuliThe experimental processwas based on a pro-totype in an experimental setting In the following sectionsthe results of the amplitude and frequency responses relativemotion high response characteristics and evaluation indicesare interpreted In addition the tuned-mass damper controlschemewill be comparedwith some existing control schemessuch as tuned-liquid damper [30] and active mass dampercontrol scheme [31]

2 Methods and Materials

21 Modeling To analyze the effectiveness of a TMD anactual jack-up offshore platform (Bohai number 5 located inthe Southern Sea) [32] was considered as the research targetAs demonstrated in Figure 1 the jack-up offshore platform


m1

m2

k1k1

k2k2

c1

xV









1119881

11989822+ 1198962(1199092minus 1199091) = minus119898

2119881

(1)

where 1198981 1198881 and 119896



2are the mass



1= 1198831119890119894119908119905 and 119909

2= 1198832119890119894119908119905



1198831=100381610038161003816100381610038161198831119890119894119908119905

10038161003816100381610038161003816=

1198981119881(1198962minus 11989821199082

)

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(2)

1198832=100381610038161003816100381610038161198832119890119894119908119905

10038161003816100381610038161003816=

11989821198811198962

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(3)



1198831

119883st

10038161003816100381610038161003816100381610038161003816

= radic(1205782

minus 1205822

)2


+ [21205851120582 (1205782 minus 1205822)]

2

(4)



1=

radic11989611198981and 119908









(119894)

=119909(119894+1)

minus 2119909(119894)

+ 119909(119894minus1)

Δ2119905

(119894)

=119909(119894+1)

minus 119909(119894minus1)

2Δ119905

(5)



1199091

(119894+1)

=

(21198981minus 1198961Δ2

119905 minus 1198962Δ2

119905) 1199091

(119894)

+ 1198962Δ2

1199051199092

(119894)

+ (minus1198981+ 05119888

1Δ119905) 1199091

(119894minus1)

minus 1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(119894+1)

=

(21198982minus 1198962Δ2

119905) 1199092

(119894)

+ 1198962Δ2

1199051199091

(119894)

minus 11989821199092

(119894minus1)

minus 1198982119881Δ2

119905

1198982

(6)


1199091

(0)

= 0

1

(0)

= 0

1

(0)

= 0

1199092

(0)

= 0

2

(0)

= 0

2

(0)

= 0

(7)


1199091

(1)

= 0

1199092

(1)

= 0

(8)


(0)

= 0 1199092

(0)

= 0 1199091

(1)

= 0 and 1199092

(1)


1

(2) and 1199092

(2)


1

(119894+1) and 1199092


1

(119894minus1) 1199092

(119894minus1) 1199091

(119894) and 1199092

(119894)


RMS = radic1

119899

119899

sum

119894=1

1199102

119894 (9)





(10)



119894 Then a





3 Experiment









TMD



Water

Sand

PC

Acceleration sensor


Platform

TMD

Shaker

Tank


Spring

Track Wheel Frame

Mass




0

2

4

6

8

10

12

0 1 2 3 4

TMDPlatform

Frequency (Hz)

Freq

uenc

y re

spon

se fu

nctio

n (d

B)


4 Results






minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)





minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)


135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)









0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










m1

m2

k1k1

k2k2

c1

xV









1119881

11989822+ 1198962(1199092minus 1199091) = minus119898

2119881

(1)

where 1198981 1198881 and 119896



2are the mass



1= 1198831119890119894119908119905 and 119909

2= 1198832119890119894119908119905



1198831=100381610038161003816100381610038161198831119890119894119908119905

10038161003816100381610038161003816=

1198981119881(1198962minus 11989821199082

)

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(2)

1198832=100381610038161003816100381610038161198832119890119894119908119905

10038161003816100381610038161003816=

11989821198811198962

radic1198881

21199082 (1198962minus 11989821199082)2

+ [(1198961+ 1198962minus 11989811199082) (119896

2minus 11989821199082) minus 119896

2

2

]2

(3)



1198831

119883st

10038161003816100381610038161003816100381610038161003816

= radic(1205782

minus 1205822

)2


+ [21205851120582 (1205782 minus 1205822)]

2

(4)



1=

radic11989611198981and 119908









(119894)

=119909(119894+1)

minus 2119909(119894)

+ 119909(119894minus1)

Δ2119905

(119894)

=119909(119894+1)

minus 119909(119894minus1)

2Δ119905

(5)



1199091

(119894+1)

=

(21198981minus 1198961Δ2

119905 minus 1198962Δ2

119905) 1199091

(119894)

+ 1198962Δ2

1199051199092

(119894)

+ (minus1198981+ 05119888

1Δ119905) 1199091

(119894minus1)

minus 1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(119894+1)

=

(21198982minus 1198962Δ2

119905) 1199092

(119894)

+ 1198962Δ2

1199051199091

(119894)

minus 11989821199092

(119894minus1)

minus 1198982119881Δ2

119905

1198982

(6)


1199091

(0)

= 0

1

(0)

= 0

1

(0)

= 0

1199092

(0)

= 0

2

(0)

= 0

2

(0)

= 0

(7)


1199091

(1)

= 0

1199092

(1)

= 0

(8)


(0)

= 0 1199092

(0)

= 0 1199091

(1)

= 0 and 1199092

(1)


1

(2) and 1199092

(2)


1

(119894+1) and 1199092


1

(119894minus1) 1199092

(119894minus1) 1199091

(119894) and 1199092

(119894)


RMS = radic1

119899

119899

sum

119894=1

1199102

119894 (9)





(10)



119894 Then a





3 Experiment









TMD



Water

Sand

PC

Acceleration sensor


Platform

TMD

Shaker

Tank


Spring

Track Wheel Frame

Mass




0

2

4

6

8

10

12

0 1 2 3 4

TMDPlatform

Frequency (Hz)

Freq

uenc

y re

spon

se fu

nctio

n (d

B)


4 Results






minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)





minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)


135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)









0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1199091

(119894+1)

=

(21198981minus 1198961Δ2

119905 minus 1198962Δ2

119905) 1199091

(119894)

+ 1198962Δ2

1199051199092

(119894)

+ (minus1198981+ 05119888

1Δ119905) 1199091

(119894minus1)

minus 1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(119894+1)

=

(21198982minus 1198962Δ2

119905) 1199092

(119894)

+ 1198962Δ2

1199051199091

(119894)

minus 11989821199092

(119894minus1)

minus 1198982119881Δ2

119905

1198982

(6)


1199091

(0)

= 0

1

(0)

= 0

1

(0)

= 0

1199092

(0)

= 0

2

(0)

= 0

2

(0)

= 0

(7)


1199091

(1)

= 0

1199092

(1)

= 0

(8)


(0)

= 0 1199092

(0)

= 0 1199091

(1)

= 0 and 1199092

(1)


1

(2) and 1199092

(2)


1

(119894+1) and 1199092


1

(119894minus1) 1199092

(119894minus1) 1199091

(119894) and 1199092

(119894)


RMS = radic1

119899

119899

sum

119894=1

1199102

119894 (9)





(10)



119894 Then a





3 Experiment









TMD



Water

Sand

PC

Acceleration sensor


Platform

TMD

Shaker

Tank


Spring

Track Wheel Frame

Mass




0

2

4

6

8

10

12

0 1 2 3 4

TMDPlatform

Frequency (Hz)

Freq

uenc

y re

spon

se fu

nctio

n (d

B)


4 Results






minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)





minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)


135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)









0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TMD



Water

Sand

PC

Acceleration sensor


Platform

TMD

Shaker

Tank


Spring

Track Wheel Frame

Mass




0

2

4

6

8

10

12

0 1 2 3 4

TMDPlatform

Frequency (Hz)

Freq

uenc

y re

spon

se fu

nctio

n (d

B)


4 Results






minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)





minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)


135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)









0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus40minus20

02040

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)




0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)



0102030

0 5 10 15 20 25Disp

lace

men

t (m

m)

Time (s)





minus5minus3minus1

135

0 5 10 15 20 25Time (s)


Acce

lera

tion

(mm

s2)


135

0 5 10 15 20 25Time (s)

minus5minus3minus1


Acce

lera

tion

(mm

s2)









0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

4

8

12

16



Pow

er sp

ectr

al d

ensit

y (m

2middots)


0

4

8

12

16

20



Pow

er sp

ectr

al d

ensit

y (m

2middots)








5 Discussion


0

02

04

06

08



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)


0

03

06

09

12



Pow

er sp

ectr

al d

ensit

y(m

2middots3

)






1199091

(0)

= 0

1199092

(0)

= 0

1199091

(1)

= 0

1199092

(1)

= 0

(12)


minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus50

minus30

minus10

10

30

50

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform


minus30

minus20

minus10

0

10

20

30

0 1 2 3

Disp

lace

men

t (m

m)

Time (s)

TMDPlatform




1

(2) and 1199092

(2)

can be obtained

1199091

(2)

=minus1198981119881Δ2

119905

1198981+ 05119888

1Δ119905

1199092

(2)

= minus119881Δ2

119905

(13)


(2) and 1199092


1199091

(2)

1199092

(2)

=1198981

1198981+ 05119888

1Δ119905

lt 1 (14)


2


1

(2) that is 1199092

(2)

gt

1199091



minus12

minus8

minus4

0

4

8

12

0 1 2 3Time (s)

TMDPlatform

Acce

lera

tion

(mm

s2)


0

4

8

0 1 2 3Time (s)

minus8

minus4

TMDPlatform

Acce

lera

tion

(mm

s2)











minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)


minus360

minus180

0

180

360

0 1 2 3 4 5Phas

e diff

eren

ce (d

egre

e)

Frequency (Hz)




119864119896=1

21198962(1199092minus 1199091)2

(15)



2minus






x1

x2

(a) |1199092minus 1199091| = 0

x1

x2

(b) |1199092minus 1199091| = 0



2minus1199091| = 0)





6 Conclusions



minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus40

minus20

0

20

40

0 5 10 15 20 25Relat

ive d

ispla

cem

ent (

mm

)

Time (s)


0

20

40

0 5 10 15 20 25Time (s)

minus40

minus20

Relat

ive d

ispla

cem

ent (

mm

)





Competing Interests


Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

High Response Performance of a Tuned-Mass Damper for Vibration