Research Article Parametric Study of Tuned Mass Dampers for...

Research ArticleParametric Study of Tuned Mass Dampers for Long SpanTransmission Tower-Line System under Wind Loads

Li Tian and Yujie Zeng

School of Civil Engineering Shandong University Jinan China

Correspondence should be addressed to Li Tian tianlisdueducn

Received 29 June 2016 Revised 17 September 2016 Accepted 3 October 2016

Academic Editor Pedro Galvın

Copyright copy 2016 L Tian and Y ZengThis 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

A parametric study of tuned mass dampers for a long span transmission tower-line system under wind loads is done in this paperA three-dimensional finite element model of transmission tower-line system is established by SAP2000 software to numericallyverify the effectiveness of the tuned mass damper device The wind load time history is simulated based on Kaimal spectrum bythe harmony superposition method The equations of motion of a system with tuned mass damper under wind load excitation areproposed and the schematic of tuned mass damper is introduced The effects of mass ratio frequency ratio damping ratio thechange of the sag of transmission line and the robustness of TMD are investigated respectively Results show that (1) the changeof mass ratio has a greater effect on the vibration reduction ratio than those of frequency ratio and damping ratio and the bestvibration reduction ratio of TMD is not the frequency ratio of 1 (2) the sag-span ratio has an insignificant effect on the vibrationreduction ratio of transmission tower when the change of sag-span ratio is not large and (3) the effect of ice should be consideredwhen the robustness study of TMD is carried out

1 Introduction

With the development of industry transmission tower andline are more and more important as a lifeline project inpower system Its failure may lead to the outage of powersupply [1] The transmission tower-line system is very sen-sitive to wind loads due to its large span and high rise andbeing flexible Many transmission towers are damaged due tostrong wind loads in recent investigations [2] For examplein 2002 the strong wind appeared in Liaoning province ofChina which resulted in the damage of many transmissiontowers Due to the occurrence of the strong wind in 2005the transmission towers are damaged seriously in Jiangsuprovince of China In 2009 the transmission tower wasbroken owing towind and ice in Australia Thewind-inducedcollapse of transmission towers is shown in Figure 1 It isof great importance to reduce the response of transmissiontowers and improve the reliability of transmission towersunder wind loads

A few researchers have studied the vibration control oftransmission tower-line system under wind action in recentyears Wu and Yang [3] designed an active mass driver for

installation on the upper observation deck of the 310mNanjing TV transmission tower in China in order to reducethe acceleration response under strong wind gusts Fur-thermore Wu and Yang [4] presented the linear quadraticGaussian (LQG) control strategy using acceleration feedbackto reduce the tower responses under coupled lateral torsionalmotion Battista et al [5] envisaged nonlinear pendulum-likedampers (NPLD) installed on the towers and demonstratedthe efficiency with the aid of comparisons between numericalresults obtained from the controlled and the uncontrolledsystems He et al [6] proposed a suspended mass pendulum(SMP) for vibration reduction of a transmission tower Bothnumerical and experimental results verified that the SMPwasvery effective in reducing acceleration of the tower Chenet al [7] carried out the control of wind-induced responseof transmission tower-line system using magnetorheological(MR) dampers The results demonstrated that the incor-poration of MR dampers into the transmission tower-linesystem can substantially suppress the wind-induced collapseof transmission tower In addition Chen et al [8] studied thevibration control and performance evaluation on a transmis-sion tower-line system by using friction dampers subjected

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4965056 11 pageshttpdxdoiorg10115520164965056

2 Shock and Vibration

Figure 1 Wind-induced collapse of transmission towers

North towerTangent type

South towerTangent type

Transmission lineSouth side tower

Tension typeNorth side tower

Tension type

Huanghe River

285m1118m294m

Figure 2 Schematic diagram of long span transmission tower-line system

to wind excitations The results showed the application offriction dampers with optimal parameters could significantlyreduce wind-induced responses of the transmission tower-line system Zhang et al [9] proposed the utilization of theinternal resonance feature of the spring pendulum (SP) toreduce the wind-induced vibration of a transmission towerTian and Gai [10] proposed a new type of vibration controldevice that is pounding tuned mass damper (PTMD) andPTMD was very effective in reducing the wind-inducedvibration and the vibration control performance improvedas the external wind load increases However there are fewstudies about the parametric study of vibration control forlong span transmission tower-line system under wind loads

A parametric study of tuned mass damper (TMD) forlong span transmission tower-line system under wind loads isinvestigated in this paper A three-dimensional finite elementmodel of long span transmission tower-line system is estab-lished by SAP2000 software based on a real project The windload time history is simulated by the harmony superpositionmethodThe finite element simulation of TMD is introducedThe effects of mass ratio frequency ratio and damping ratioof TMD are studied respectively Furthermore the effects ofchange of the sag of transmission line and the robustness ofTMD are investigated respectively

2 Structural Model

The long span transmission tower-line system across Huang-he River in China is studied in this paper Figure 2 givesthe schematic diagram of long span transmission tower-linesystem The long span transmission tower-line system con-sists of four transmission towers and three span transmissionlines North and south side towers are tension type while

Figure 3 Practical graph of transmission tower-line system

north and south towers are tangent type The span across theHuanghe River is 1118m and the north and south side spansare 294m and 285m respectively Figure 3 gives the practicalgraph of transmission tower-line system The size of tangenttower with the total height of 122m is shown in Figure 4 Thestructural members of the transmission tower are made ofsteel tubewith 206 Gpa elasticmodulusThe type of the uppertwo ground lines is OPGW-180 and the type of the lower sixtwo-bundled conductor lines is LHBGJ-40095 The three-dimensional finite model of the coupled transmission tower-line system is established by SAP2000 software as shown inFigure 5 The beam and cable element are selected for thesimulation of transmission tower and line respectively Thebase points of the transmission tower are assumed to be fixed

Shock and Vibration 3

25

102

105

95

122

13

18 5 18

1211109

876

5

4

3

2

1

Figure 4 Tower size (m)

TransverseLongitudinal

Vertical1

2285m

1118m

294m

Figure 5 Three-dimensional finite model of the long span trans-mission tower-line coupled system

and the connections between transmission towers and linesare hinged by insulators As shown in Figure 5 the wind angleof 0∘ can be assumed as the longitudinal direction along thetransmission line while the wind angle of 90∘ can be assumedas the transverse direction perpendicular to the transmissionline

3 Wind Load Simulation

The wind excitations of transmission tower-line system aresimulated by harmony superposition method [11 12] Thetarget power spectrum is Kaimal fluctuating wind powerspectrum [13] Wind is generated due to the air flow in theatmospheric boundary layerThemovement of the wind fieldis a heterogeneous random process related to the space andtime Wind velocity profile is used to describe the variationof the mean wind along the height The logarithmic functionis used to calculate the mean wind and can be expressed as

V (119911) = 1119896119906lowast ln(1199111199110) (1)

Table 1 The positions of wind load and windward area

Points Height (m) Windward area (m2)1 8 21662 2375 17813 385 13394 505 11625 615 8216 715 7807 805 5118 865 4159 92 57210 995 34911 1045 139312 10975 33513 1165 1566

where 119911 is the standard height 119896 is the Karman constant equalto 04 V(119911) is the mean wind speed at the height of 119911 1199110 is theroughness length 119906lowast is the friction velocity

The Kaimal spectrum is adopted as target spectrum tosimulated turbulent wind for the high transmission towerThe Kaimal spectrum can be expressed as

119899119878V (119899)119906lowast = 200119909

(1 + 50119909)53 (2)

where 119909 = 1200119899V10 V10 is the wind speed of standardheight 119899 is the frequency of fluctuating wind

The harmony superposition method proposed by Shi-nozuka is a numerical simulation method to simulate thesteady random process According to the theory of Shi-nozuka when 119873 rarr infin velocity time series of fluctuatingwind 119906119894(119905) can be expressed as follows

119906119894 (119905)

=119894

sum119897=1

119873

sum119896=1

1003816100381610038161003816119867119894119897 (120596119896)1003816100381610038161003816 radic2Δ120596119896 cos [120596119896119905 minus 120579119894119897 (120596119896) + 120593119897119896]

119894 = 1 2 119898

(3)

where 119873 is the division numbers of fluctuating wind fre-quency 119867119894119897 is obtained from Cholesky decomposition of thewind cross-spectral density matrix 120579119894119897(120596) is the phase anglebetween the two different loading points 120593119897119896 is the uniformlyrandom numbers in [0 2120587] Δ120596119896 is the frequency increment120596119896 = 119896Δ120596119896 minus ((119873 minus 119897)119873)Δ120596119896

It is very difficult to simulate the wind velocity historyfor every node of the transmission tower and line becausethe nodes of the transmission tower and line are too muchConsidering sufficient accuracy the transmission tower isdivided into 13 regions and each region can be simplified asone simulation point shown in Figure 4The positions of eachregion center and windward area are listed in Table 1 Themiddle span and side span of transmission line are dividedinto 27 and 12 regions respectively The spatial correlationfunction for wind velocities is taken into account in the


Point 5

50 100 150 200 250 3000Time (s)

0

10

20

30

40W

ind

spee

d (m

s)

minus10

minus20

minus30

(a) Simulation point 5

0

10

20

30

Win

d sp

eed

(ms

)

Point 10

minus10

minus20

minus30

50 100 150 200 250 3000Time (s)

(b) Simulation point 10

Figure 6 The simulated fluctuating wind time histories of points 5 and 10

Frequency (Hz)Simulated spectrumTarget spectrum

Pow

er sp

ectr

um d

ensit

y (m

2s

3)

10010minus110minus210minus310minus3

10minus2

10minus1

100

101

102

103

104


Simulated spectrumTarget spectrum

Frequency (Hz)10010minus110minus210minus3

Pow

er sp

ectr

um d

ensit

y (m

2s

3)

10minus3

10minus2

10minus1

100

101

102

103

104


Figure 7 Comparison between simulated and objective wind spectra

mathematical model for wind forces acting on lines andtowers

Mean wind speed at 10m height is assumed to be 30msBased on the above method the fluctuating wind speeds ofwind loads are simulated The fluctuating wind speed timehistories of points 5 and 10 are shown in Figure 6 Thecomparison between simulated and target spectra is to verifythe reliability of the simulation method as shown in Figure 7It can be seen from the figure that the spectral line trend ofsimulated spectrum is consistent with that of target spectrumwhich illustrates the simulation method is reasonable

The along-wind loads are calculated by the following

119865 (119905) = 120583119904119860119881 (119905)2

16 (4)

where 119860 is the windward area in Table 1 120583119904 is the shapecoefficient The shape coefficients of transmission tower andline are equal to 135 and 11 respectively 119881(119905) is the windspeed of simulation point

4 Mechanism of the TMD

41 Equations of Motion of a System with TMD underWind Load TMD is a dynamic absorber consisting of amass spring and damping The TMD is considered in thetransmission tower only in the paper The structure withtuned mass damper can be simplified as a single-degree-of-freedom system as shown inFigure 8The equation ofmotionof TMD can be written as

119898119889 (119889 + ) + 119888119889119889 + 119896119889119909119889 = 0 (5)


K

M

C

kd

md

cd

F(t)

Figure 8 Schematic structure with TMD

kd

md

cd

kmcm

Figure 9 Schematic simulation of TMD

where 119909119889 is the displacement of the TMD and 119909 is thedisplacement of the structure 119898119889 119888119889 and 119896119889 are the massdamping and stiffness of the TMD respectively The massdamping and stiffness of the TMD would be calculated inthe next section

The equation of motion of structure with TMD underwind load can be expressed as

119872 + 119862 + 119870119909 minus 119888119889119889 minus 119896119889119909119889 = 119865 (119905) (6)

where119872 119862 and 119870 are the mass damping and stiffness ofthe structure respectively The mass damping and stiffnessof the structure are calculated by the along-wind fundamentalvibration mode for wind directions 0∘ and 90∘ respectively119865(119905) is the wind load

42 Finite Element Simulation of TMD The TMD is simu-lated by mass element linear element and damping elementin SAP2000 as shown in Figure 9 The mass stiffness anddamping of TMD can be derived as

119898119889 = 120583119872119896119889 = 1198981198891205962119889119888119889 = 2119898119889120596119889120585

(7)

where119872 is the mass of the control structure 120583 is the massratio 120596119889 is the frequency of the control structure 120585 is thedamping ratio of the TMD

TheMaxwell calculation model of damping 119888119898 and spring119896119898 in series is used to simulate damping 119888119889 as shownin Figure 10 [14] The displacement of the spring 119896119898 and

i j

kmcm

Figure 10 Schematic of Maxwell model

damping 119888119898 is defined as 119889119896 and 119889119888 respectively The relatedequations of the Maxwell model can be expressed as

119891119889 = 119896119898119889119896 = 119888119898119888119889 = 119889119896 + 119889119888

(8)

where 119889 is the deformation between 119894 and 119895 When 119896119898 is largeenough and 119889119896 is small enough 119889 = 119889119888 can be obtainedMeanwhile 119891119889 = 119888119898 can be derived and 119888119898 is the dampingof TMD

5 Parameters Study of TMD

Thetransmission tower-line systemmodels shown inFigure 4without control and with the TMD are analyzed respectivelyThe Newmark-120573method is applied in the numerical integra-tion The geometric nonlinearity is taken into account due tolarge deformation The damping ratios of the transmissiontower and transmission line are assumed to be 002 and001 respectively The vibration of the first mode of thetransmission tower under wind load is controlled by TMDand the TMD is installed on the top of the transmission towerTo obtain the optimal control parameters the effects of massratio frequency ratio and damping ratio are investigatedrespectively The mass ratio between TMD and structure canbe defined as 120583 = 119898119889119872 times 100 The frequency ratiobetween TMD and structure can be defined as 119891 = 120596119889120596119904The vibration reduction ratio can be defined as

120575 = 1198770 minus 11987711198770 times 100 (9)

where 1198770 and 1198771 are the response of the transmission towerwithout and with control respectively

51 Effect of Mass Ratio To study the effect of mass ratiobetween TMD and tower on the vibration reduction ratioeight different mass ratios are considered in the analysis05 1 2 3 4 5 6 and 7 to cover the rangeof the mass ratio in the engineering In all these cases thefrequency ratio between TMD and tower is assumed to be 1and the damping ratio of TMD is assumed to be 01

As shown in Figure 11 the vibration reduction ratios ofaxial force along the height of tower with different mass ratiosare given It can be seen from the figure that the vibrationreduction ratio of axial force of tower under different windattack angles increases gradually with the increasing massratio The maximum vibration reduction ratio of axial forceof tower could exceed 50 when the mass ratio reaches 7The vibration reduction ratios of axial force of tower under 0∘wind attack angle are less than those under 90∘ wind attackangle when the mass ratio is less than 4 However thevibration reduction ratios of axial force of tower under 0∘


05123

4567

0

50

100

150H

eigh

t (m

)

10 20 30 40 50 600Vibration reduction rate

(a) 0∘ wind attack angle

0

50

100

150

Hei

ght (

m)


05123

4567

(b) 90∘ wind attack angle

Figure 11 Vibration reduction ratio of axial force along the height of tower

Displacement Acceleration

2 4 6 80Mass ratio ()

0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



Figure 12 Vibration reduction ratios of displacement and acceleration at the top of tower

wind attack angle are close to those under 90∘ wind attackangle when the mass ratio is larger than 5 The results alsoshow that the increasing speed of the vibration reductionratio of axial force decreases with the increasing mass ratio

The vibration reduction ratios of displacement and accel-eration at the top of tower with different mass ratios underdifferent wind attack angles are shown in Figure 12 Asshown in Figure 12(a) the vibration reduction ratios ofdisplacement and acceleration under 0∘ wind attack anglehave an increasing tendency with the increasing mass ratioHowever the increasing speed of the vibration reductionratio of acceleration obviously decreases when the mass ratio

exceeds 4 The vibration reduction ratio of displacementis not significant with the increasing mass ratio It alsocan be seen from Figure 12(b) that the increasing speedsof the vibration reduction ratios of the displacement andacceleration under 90∘ wind attack angle are slow whenthe mass ratio exceeds 3 The vibration reduction ratio ofacceleration decreases with the increasing mass ratio whenthe mass ratio exceeds 5 and the curve of the vibrationreduction ratio of acceleration shows inflection point

The above analysis shows that the increasing mass ratiohas a significant effect on the vibration reduction ratio Theincreasing speed of the vibration reduction ratio of tower


090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)

10 20 30 40 500Vibration reduction rate


0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110



under 0∘ wind attack angle decreases when the mass ratioexceeds 4 so the mass ratio of 4 could be selected as anoptimal mass ratio The increasing speeds of the vibrationreduction ratios of tower under 90∘ wind attack angle are slowwhen the mass ratio exceeds 3 and the mass ratio of 3could be selected as an optimal mass ratio

52 Effect of Frequency Ratio To investigate the effect offrequency ratio between TMD and tower on the vibrationreduction ratio eleven different frequency ratios are consid-ered in the analysis 090 092 094 096 098 100 102 104106 108 and 110 In all these cases the mass ratio betweenTMD and tower is assumed to be 2 and the damping ratioof TMD is assumed to be 01

The vibration reduction ratios of axial force along theheight of tower with different frequency ratios under differentwind attack angles are shown in Figure 13 It is clear fromFigure 13 that the vibration reduction ratio of axial forcedecreases gradually with the increasing frequency ratio Italso can be seen from the figure that the change of frequencyratio has a little influence on the vibration reduction ratioof axial force under 90∘ wind attack angle The minimumvibration reduction ratio of axial force exceeds 10 with thevariation of frequency ratios

The vibration reduction ratios of displacement and accel-eration at the top of tower with different frequency ratiosunder different wind attack angles are shown in Figure 14It is clear that the vibration reduction ratios of accelerationincrease with the increasing frequency ratio However thevariation tendencies of vibration reduction ratio of dis-placement under two wind attack angles are different Thevibration reduction ratio of displacement under 0∘ wind

attack angle decreaseswith the increasing frequency ratio butthe vibration reduction ratio of displacement under 90∘ windattack angle increases with the increasing frequency ratio

From the analysis above it can be seen that the changeof frequency ratio has a little influence on the vibrationreduction ratio Considering the vibration reduction ratioof axial force along the height of tower displacement andacceleration at the top of tower the frequency ratio of 098could be selected as an optimal frequency ratio

53 Effect of Damping Ratio To research the effect of damp-ing ratio of TMD on the vibration reduction ratio elevendifferent damping ratios are considered in the analysis 0002 004 006 008 010 012 014 016 018 and 020 Inall these cases the mass ratio and frequency ratio betweenTMD and tower are assumed to 2 and 1 respectively

The vibration reduction ratios of axial force along theheight of tower with different damping ratios are shownin Figure 15 It is clear from the figure that the vibrationreduction ratio of axial force decreases with the increasingdamping ratio The variations of damping ratio have a greaterinfluence on the vibration reduction ratio of axial force under0∘ wind attack angle than that of under 90∘ wind attack angle

Figure 16 shows the vibration reduction ratios of displace-ment and acceleration at the top of tower under different windattack angles The vibration reduction ratio of accelerationdecreases gradually with the increasing damping ratio whenthe damping ratio exceeds 002 The variation of vibrationreduction ratio of displacement under 0∘ wind attack angleis just opposite to that of under 90∘ wind attack angle Thevibration reduction ratio of displacement under 0∘ windattack angle decreases gradually with the increasing damping



092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio


Figure 14 Vibration reduction ratio of displacement and acceleration at the top of tower

10 20 30 400Vibration reduction rate

0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)

10 20 30 40 500Vibration reduction rate ()

0002004006008010

012014016018020



ratio but the vibration reduction ratio of displacement under90∘ wind attack angle increases gradually with the increasingdamping ratio

The above discussion shows that the damping ratio hasa greater influence on the vibration reduction ratio under 0∘wind attack angle than that of under 90∘ wind attack angleConsidering the vibration reduction ratio of axial force alongthe height of tower displacement and acceleration at the topof tower the optimal damping ratio of 008 could be selectedas an optimal damping ratio

In conclusion the mass ratio has a greater effect on thevibration reduction ratio than those of frequency ratio anddamping ratio The best vibration reduction ratio of TMD isnot the frequency ratio of 1 The optimal parameters of TMDunder different wind attack angles are listed in Table 2

6 Effect of Sag-Span Ratio

In this section the change of the sag of transmission line isdiscussed based on the optimal parameters of TMD shown



0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te

minus002 006 010 014 018 022002Damping ratio




0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate



Table 2 Optimal parameters of TMD under different wind attackangles

Optimal parameter 0∘ wind attack angle 90∘ wind attack angleMass ratio () 3 4Frequency ratio 098 098Damping ratio 008 008

in Table 2 The current state of the sag of transmission lineis 55m and the span of transmission towers is 1118m so thesag-span ratio is 0049Thevibration reduction ratios of TMDunder different wind attack angles considering the effectof sag-span ratio are analyzed respectively The vibrationreduction ratios of TMD with different sag-span ratio arelisted inTable 3 It can be seen from the table that the sag-spanratio has a little influence on the vibration reduction ratioBecause the span of the transmission towers is very large andthe change of the sag of transmission line ranges in 5 metersthe change of sag-span ratio is small Therefore the sag-spanratio has an insignificant effect on the vibration reductionratio of transmission tower when the change of sag-span ratiois not large

7 Robustness Study

To study the robustness of TMD the case that ice loadsare on the transmission tower and line is taken into con-sideration The increasing of the cross sectional area ofthe members insulators and transmission lines caused byicing is equivalent to the change of the material density intransmission tower-line system Meanwhile the wind loadsin each simulation point are varied owing to the increasing ofthe windshield area The optimal parameters of TMD underdifferent wind attack angles shown in Table 2 are selected inthis section

As shown in Table 4 the vibration reduction ratios ofTMDof the transmission tower-line systemwithout and withconsidering ice are analyzed respectively The thickness ofice is assumed to be 5mm It can be seen from Table 4 thatthe vibration reduction ratios of axial force and accelerationdecrease when the ice load is considered but the vibrationreduction ratio of the displacement increases Owing to theeffect of ice the vibration frequency and wind load of thetransmission tower-line system should be changed and itwould affect the vibration reduction ratio Therefore theeffect of ice should be considered when the robustness studyof TMD is carried out

8 Conclusion

The parametric study of tuned mass damper for long spantransmission tower-line system under wind loads is carriedout The three-dimensional finite element model of longspan transmission tower-line system is established The windload time history is simulated by the harmony superpositionmethod The effects of mass ratio frequency ratio dampingratio the sag of transmission line and the robustness ofTMD are investigated respectively Based on above study thefollowing conclusions are drawn

(1) The increasing mass ratio has a significant effect onthe vibration reduction ratio The vibration reductionratios increase with the increasing mass ratio but theincreasing speed of the vibration reduction ratio oftower decreases when themass ratio exceeds a certainratio The mass ratio of 4 and 3 could be selectedas an optimal mass ratio under 0∘ and 90∘ wind attackangles respectively

(2) The change of the frequency ratio has a little influenceon the vibration reduction ratio The vibration reduc-tion ratios of axial force and acceleration increase


Table 3 Vibration reduction ratio of TMD with different sag-span ratio ()

Response Line sag (m) Sag-span ratio Wind angle of 90∘ Wind angle of 0∘

Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634

Table 4 Vibration reduction ratio of TMD without and with considering ice ()

Response Ice thickness (mm) Wind angle of 90∘ Wind angle of 0∘

Axial force 0 172 2835 168 224

Displacement 0 38 625 67 71

Acceleration 0 502 6135 44 58

with the increasing frequency ratio but the variationtendencies of vibration reduction ratio of displace-ment under two wind attack angles are different Thefrequency ratio of 098 could be selected as an optimalfrequency ratio under 0∘ and 90∘ wind attack angles

(3) The damping ratio has a greater influence on thevibration reduction ratio under 0∘ wind attack anglethan that of under 90∘ wind attack angle The vibra-tion reduction ratios of axial force and accelerationdecrease with the increasing damping ratio while thevariation of vibration reduction ratio of displacementunder 0∘ wind attack angle is just opposite to that ofunder 90∘ wind attack angle The damping ratio of008 could be selected as an optimal damping ratiounder 0∘ and 90∘ wind attack angles

(4) The sag-span ratio has an insignificant effect on thevibration reduction ratio of transmission tower whenthe change of sag-span ratio is not large The changeof vibration frequency and wind load of the transmis-sion tower-line system should be varied due to theeffect of ice and the vibration reduction ratio wouldbe changed The effect of ice should be consideredwhen the robustness study of TMD is carried out

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by PhD Programs Foundationof Ministry of Education of China Project under Grant no

20120131120036 and Shandong Provincial Natural ScienceFoundation of China under Grant no ZR 2012EEQ005

References

[1] H-N Li W-L Shi G-X Wang and L-G Jia ldquoSimplifiedmodels and experimental verification for coupled transmissiontower-line system to seismic excitationsrdquo Journal of Sound andVibration vol 286 no 3 pp 569ndash585 2005

[2] J-H Park B-W Moon K-W Min S-K Lee and C KKim ldquoCyclic loading test of friction-type reinforcing membersupgrading wind-resistant performance of transmission towersrdquoEngineering Structures vol 29 no 11 pp 3185ndash3196 2007

[3] J C Wu and J N Yang ldquoActive control of transmission towerunder stochastic windrdquo Journal of Structural Engineering vol124 no 11 pp 1302ndash1312 1998

[4] J-C Wu and J N Yang ldquoLQG control of lateral-torsionalmotion of Nanjing TV transmission towerrdquo Earthquake Engi-neering and Structural Dynamics vol 29 no 8 pp 1111ndash11302000

[5] R C Battista R S Rodrigues andM S Pfeil ldquoDynamic behav-ior and stability of transmission line towers under wind forcesrdquoJournal of Wind Engineering and Industrial Aerodynamics vol91 no 8 pp 1051ndash1067 2003

[6] Y-F He W-J Lou B-N Sun S-C Yu Y Guo and Y YeldquoWind-induced vibration control of long span transmissiontower with suspended mass pendulumsrdquo Journal of ZhejiangUniversity (Engineering Science) vol 39 no 12 pp 1891ndash18962005

[7] B Chen J Zheng and W L Qu ldquoControl of wind-inducedresponse of transmission tower-line system by using mag-netorheological dampersrdquo International Journal of StructuralStability and Dynamics vol 9 no 4 pp 661ndash685 2009

[8] B Chen X Xiao P-Y Li and W-L Zhong ldquoPerformanceevaluation on transmission tower-line system with passive


friction dampers subjected to wind excitationsrdquo Shock andVibration vol 2015 Article ID 310458 13 pages 2015

[9] P Zhang L Ren H Li Z Jia and T Jiang ldquoControl of wind-induced vibration of transmission tower-line system by using aspring pendulumrdquo Mathematical Problems in Engineering vol2015 Article ID 671632 10 pages 2015

[10] L Tian and X Gai ldquoWind-induced vibration control of powertransmission tower using pounding tuned mass damperrdquo Jour-nal of Vibroengineering vol 17 no 7 pp 3693ndash3701 2015

[11] M Shinozuka ldquoSimulation of multivariate and multidimen-sional random processesrdquo The Journal of the Acoustical Societyof America vol 49 no 1 pp 357ndash368 1971

[12] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[13] J C Kaimal J C Wyngaard Y Izumi and O R Cote ldquoSpectralcharacteristics of surface-layer turbulencerdquoQuarterly Journal ofthe Royal Meteorological Society vol 98 no 417 pp 563ndash5891972

[14] R Rana and T T Soong ldquoParametric study and simplifieddesign of tuned mass dampersrdquo Engineering Structures vol 20no 3 pp 193ndash204 1998

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Figure 1 Wind-induced collapse of transmission towers

North towerTangent type

South towerTangent type

Transmission lineSouth side tower

Tension typeNorth side tower

Tension type

Huanghe River

285m1118m294m

Figure 2 Schematic diagram of long span transmission tower-line system

to wind excitations The results showed the application offriction dampers with optimal parameters could significantlyreduce wind-induced responses of the transmission tower-line system Zhang et al [9] proposed the utilization of theinternal resonance feature of the spring pendulum (SP) toreduce the wind-induced vibration of a transmission towerTian and Gai [10] proposed a new type of vibration controldevice that is pounding tuned mass damper (PTMD) andPTMD was very effective in reducing the wind-inducedvibration and the vibration control performance improvedas the external wind load increases However there are fewstudies about the parametric study of vibration control forlong span transmission tower-line system under wind loads

A parametric study of tuned mass damper (TMD) forlong span transmission tower-line system under wind loads isinvestigated in this paper A three-dimensional finite elementmodel of long span transmission tower-line system is estab-lished by SAP2000 software based on a real project The windload time history is simulated by the harmony superpositionmethodThe finite element simulation of TMD is introducedThe effects of mass ratio frequency ratio and damping ratioof TMD are studied respectively Furthermore the effects ofchange of the sag of transmission line and the robustness ofTMD are investigated respectively

2 Structural Model

The long span transmission tower-line system across Huang-he River in China is studied in this paper Figure 2 givesthe schematic diagram of long span transmission tower-linesystem The long span transmission tower-line system con-sists of four transmission towers and three span transmissionlines North and south side towers are tension type while

Figure 3 Practical graph of transmission tower-line system

north and south towers are tangent type The span across theHuanghe River is 1118m and the north and south side spansare 294m and 285m respectively Figure 3 gives the practicalgraph of transmission tower-line system The size of tangenttower with the total height of 122m is shown in Figure 4 Thestructural members of the transmission tower are made ofsteel tubewith 206 Gpa elasticmodulusThe type of the uppertwo ground lines is OPGW-180 and the type of the lower sixtwo-bundled conductor lines is LHBGJ-40095 The three-dimensional finite model of the coupled transmission tower-line system is established by SAP2000 software as shown inFigure 5 The beam and cable element are selected for thesimulation of transmission tower and line respectively Thebase points of the transmission tower are assumed to be fixed


25

102

105

95

122

13

18 5 18

1211109

876

5

4

3

2

1



Vertical1

2285m

1118m

294m





V (119911) = 1119896119906lowast ln(1199111199110) (1)





119899119878V (119899)119906lowast = 200119909

(1 + 50119909)53 (2)



119906119894 (119905)

=119894

sum119897=1

119873

sum119896=1

1003816100381610038161003816119867119894119897 (120596119896)1003816100381610038161003816 radic2Δ120596119896 cos [120596119896119905 minus 120579119894119897 (120596119896) + 120593119897119896]

119894 = 1 2 119898

(3)




Point 5

50 100 150 200 250 3000Time (s)

0

10

20

30

40W

ind

spee

d (m

s)

minus10

minus20

minus30


0

10

20

30

Win

d sp

eed

(ms

)

Point 10

minus10

minus20

minus30

50 100 150 200 250 3000Time (s)




Pow

er sp

ectr

um d

ensit

y (m

2s

3)


10minus2

10minus1

100

101

102

103

104




Pow

er sp

ectr

um d

ensit

y (m

2s

3)

10minus3

10minus2

10minus1

100

101

102

103

104






119865 (119905) = 120583119904119860119881 (119905)2

16 (4)




119898119889 (119889 + ) + 119888119889119889 + 119896119889119909119889 = 0 (5)


K

M

C

kd

md

cd

F(t)


kd

md

cd

kmcm




119872 + 119862 + 119870119909 minus 119888119889119889 minus 119896119889119909119889 = 119865 (119905) (6)



119898119889 = 120583119872119896119889 = 1198981198891205962119889119888119889 = 2119898119889120596119889120585

(7)



i j

kmcm



119891119889 = 119896119898119889119896 = 119888119898119888119889 = 119889119896 + 119889119888

(8)




120575 = 1198770 minus 11987711198770 times 100 (9)





05123

4567

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


05123

4567





0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate









090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










25

102

105

95

122

13

18 5 18

1211109

876

5

4

3

2

1



Vertical1

2285m

1118m

294m





V (119911) = 1119896119906lowast ln(1199111199110) (1)





119899119878V (119899)119906lowast = 200119909

(1 + 50119909)53 (2)



119906119894 (119905)

=119894

sum119897=1

119873

sum119896=1

1003816100381610038161003816119867119894119897 (120596119896)1003816100381610038161003816 radic2Δ120596119896 cos [120596119896119905 minus 120579119894119897 (120596119896) + 120593119897119896]

119894 = 1 2 119898

(3)




Point 5

50 100 150 200 250 3000Time (s)

0

10

20

30

40W

ind

spee

d (m

s)

minus10

minus20

minus30


0

10

20

30

Win

d sp

eed

(ms

)

Point 10

minus10

minus20

minus30

50 100 150 200 250 3000Time (s)




Pow

er sp

ectr

um d

ensit

y (m

2s

3)


10minus2

10minus1

100

101

102

103

104




Pow

er sp

ectr

um d

ensit

y (m

2s

3)

10minus3

10minus2

10minus1

100

101

102

103

104






119865 (119905) = 120583119904119860119881 (119905)2

16 (4)




119898119889 (119889 + ) + 119888119889119889 + 119896119889119909119889 = 0 (5)


K

M

C

kd

md

cd

F(t)


kd

md

cd

kmcm




119872 + 119862 + 119870119909 minus 119888119889119889 minus 119896119889119909119889 = 119865 (119905) (6)



119898119889 = 120583119872119896119889 = 1198981198891205962119889119888119889 = 2119898119889120596119889120585

(7)



i j

kmcm



119891119889 = 119896119898119889119896 = 119888119898119888119889 = 119889119896 + 119889119888

(8)




120575 = 1198770 minus 11987711198770 times 100 (9)





05123

4567

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


05123

4567





0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate









090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Point 5

50 100 150 200 250 3000Time (s)

0

10

20

30

40W

ind

spee

d (m

s)

minus10

minus20

minus30


0

10

20

30

Win

d sp

eed

(ms

)

Point 10

minus10

minus20

minus30

50 100 150 200 250 3000Time (s)




Pow

er sp

ectr

um d

ensit

y (m

2s

3)


10minus2

10minus1

100

101

102

103

104




Pow

er sp

ectr

um d

ensit

y (m

2s

3)

10minus3

10minus2

10minus1

100

101

102

103

104






119865 (119905) = 120583119904119860119881 (119905)2

16 (4)




119898119889 (119889 + ) + 119888119889119889 + 119896119889119909119889 = 0 (5)


K

M

C

kd

md

cd

F(t)


kd

md

cd

kmcm




119872 + 119862 + 119870119909 minus 119888119889119889 minus 119896119889119909119889 = 119865 (119905) (6)



119898119889 = 120583119872119896119889 = 1198981198891205962119889119888119889 = 2119898119889120596119889120585

(7)



i j

kmcm



119891119889 = 119896119898119889119896 = 119888119898119888119889 = 119889119896 + 119889119888

(8)




120575 = 1198770 minus 11987711198770 times 100 (9)





05123

4567

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


05123

4567





0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate









090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










K

M

C

kd

md

cd

F(t)


kd

md

cd

kmcm




119872 + 119862 + 119870119909 minus 119888119889119889 minus 119896119889119909119889 = 119865 (119905) (6)



119898119889 = 120583119872119896119889 = 1198981198891205962119889119888119889 = 2119898119889120596119889120585

(7)



i j

kmcm



119891119889 = 119896119898119889119896 = 119888119898119888119889 = 119889119896 + 119889119888

(8)




120575 = 1198770 minus 11987711198770 times 100 (9)





05123

4567

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


05123

4567





0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate









090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










05123

4567

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


05123

4567





0

20

40

60

80

Vibr

atio

n re

duct

ion

rate



0

20

40

60

80

Vibr

atio

n re

duct

ion

rate









090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










090092094096098100

102104106108110

0

50

100

150H

eigh

t (m

)



0

50

100

150

Hei

ght (

m)


090092094096098100

102104106108110














092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











092 096 100 104 108 112088Frequency ratio

0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te



0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate

092 096 100 104 108 112088Frequency ratio




0

50

100

150

Hei

ght (

m)

0002004006008010

012014016018020


0

50

100

150

Hei

ght (

m)


0002004006008010

012014016018020










0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

10

20

30

40

50Vi

brat

ion

redu

ctio

n ra

te





0

10

20

30

40

50

Vibr

atio

n re

duct

ion

rate






7 Robustness Study



8 Conclusion







Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






Competing Interests


Acknowledgments



References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Axial force50 0045 196 29755 0049 172 28360 0054 191 301

Displacement50 0045 42 5755 0049 38 6260 0054 46 56

Acceleration50 0045 461 60155 0049 502 61360 0054 456 634



Axial force 0 172 2835 168 224






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









Research Article Parametric Study of Tuned Mass Dampers for...

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