Research ArticlePerformance Evaluation on Transmission Tower-Line Systemwith Passive Friction Dampers Subjected to Wind Excitations

Bo Chen1 Xiang Xiao2 Peng-yun Li3 and Wan-li Zhong3

1 Key Laboratory of Roadway Bridge and Structural Engineering Wuhan University of Technology Mail Box No 219No 122 Luoshi Road Wuhan 430070 China

2 School of Transportation Wuhan University of Technology Wuhan 430070 China3 Guangdong Power Grid Corporation Co Ltd Guangzhou 510080 China

Correspondence should be addressed to Bo Chen cbsteven163com

Received 30 August 2014 Accepted 21 October 2014

Academic Editor Ting-Hua Yi

Copyright copy 2015 Bo Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The vibration control and performance evaluation on a transmission-tower line system by using friction dampers subjected towind excitations are carried out in this study The three-dimensional finite element (FE) model of a transmission tower is firstlyconstructed A two-dimensional lumped mass model of a transmission tower is developed for dynamic analysis The analyticalmodel of transmission tower-line system is proposed by taking the dynamic interaction between the tower and the transmissionlines into consideration The mechanical model of passive friction damper is presented by involving the effects of damper axialstiffness The equation of motion of the transmission tower-line system incorporated with the friction dampers disturbed by windexcitations is established A real transmission tower-line system is taken as an example to examine the feasibility and reliability ofthe proposed control approach An extensive parameter study is carried out to find the optimal parameters of friction damper andto assess the effects of slipping force axial stiffness and hysteresis loop on control performance The work on an example structureindicates that the application of friction dampers with optimal parameters could significantly reduce wind-induced responses ofthe transmission tower-line system

1 Introduction

Overhead transmission tower-line system is a typical kindof electrical power infrastructures widely used throughoutthe world for energy supplying To be a high-rise structurethe transmission tower-line system is prone to strong windexcitations due to its small damping [1ndash3] The excessivevibration of a transmission tower-line system due to strongwind loadings may induce the structural damage or failureassociated with the events such as member fracture memberbuckling and tower collapse The failure of the towers underwind loading has been frequently reported across the world[4ndash6] Tomitigate the dynamic responses of the transmissiontower-line system many theoretical experimental and fieldmeasurement investigations have been carried out duringthe past two decades [7ndash9] The current approaches andtechniques used for performance evaluation and controlcan be classified into two major categories The first one

is the conventional approach developed by increasing thestructural stiffness while accepting an intensive level ofwind excitations The other one is an alternative approachto prevent structural failure by installing vibration controldevices

Current studies on the vibration control of transmissiontower-line systems focus on the application of dynamicabsorbers and energy-dissipating dampers [10] A typicaldevice commonly utilized for vibration mitigation of thehigh-rise structures is the tuned mass damper (TMD) Theapplication of the TMD can reduce the structural dynamicresponses to some extent while several additional massesshould be installed on top of a transmission tower whichrequires the occupancy of the structural space In additiondue to the passive nature of TMD it can only suppressthe vibration of tuned mode shapes instead of the globaldynamic responses Therefore the control performance ofthe TMD on transmission tower-line system is limited To

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 310458 13 pageshttpdxdoiorg1011552015310458

2 Shock and Vibration

overcome the shortcoming of the dynamic absorbers manyenergy-dissipating dampers have been developed recentlyas an alternative approach for dynamic mitigation of trans-mission tower-line system Up to now the vibration controlof transmission tower-line system subjected to dynamicexcitations has been investigated rarely [11 12] Chen et alproposed a new method for the wind-resistant design of thetransmission tower-line system by using viscoelastic dampers[13] Chen et al also proposed a novel approach for the wind-induced semiactive vibration control of transmission tower-line system by using magnetorheological (MR) dampersHowever it is widely reported that the performance of theviscoelastic dampers can be substantially weakened underthe harsh environment which cannot be avoided for thetransmission tower-line system in the open air [10 14]The configuration and fabrication of the semiactive controldevices such as MR damper are quite complicated andthe requirement in the additional energy supply duringthe vibration control process is unrealistic while acceptingstrong excitations Therefore the passive friction damperswith a simple configuration and excellent environmentaladaptability can be adopted to suppress the wind-inducedresponses of the transmission tower-line systemThe frictiondamper is a typical energy-dissipating device utilized in thestructural response control application and it can be manu-factured as an axial member to replace common structuralmembers of a tower without requiring the additional spaceoccupancy

To this end the vibration control and performanceevaluation on a transmission tower-line system with passivefriction dampers subjected to wind loading is actively carriedout in this studyThe three-dimensional finite element modelof a transmission tower is firstly constructed and then a two-dimensional lumped mass model is also developed The ana-lytical model of transmission tower-line system is proposedby taking the dynamic interaction between the tower andthe transmission lines into consideration The mechanicalmodel of the friction damper is presented by consideringthe effects of damper axial stiffness The equation of motionof the transmission tower-line system incorporated with thefriction dampers disturbed by wind excitations is establishedfor both the in-plane and out-of-plane vibration respec-tively A real transmission tower-line system constructedin southern China is taken as an example to examine thefeasibility and reliability of the proposed control approach Inaddition a parametric study is conducted in detail in orderto examine the effects of the slipping force the axial stiffnessand the hysteresis loop on the control performanceThemadeobservations indicate that the passive friction dampers can beutilized to suppress the wind-induced vibration control of thetransmission tower-line system due to its satisfactory energy-dissipating capacity if the damper parameters are optimallydetermined

2 Model of Transmission Tower-Line System

21 Model of Transmission Tower To be a typical truss towerconstructed by using steelmembers a transmission tower can

9000

9800

7500

8000

107800

17000

12500

12500

98000

15000

13800

11200

12500

13000

(a) Three-dimensional FE model

107800

13000

25500

36700

50500

65500

78000

90500

980009

8

7

6

5

4

3

2

1

(b) Lumped massmodel

Figure 1 Analytical model of a transmission tower

be commonlymodelled by using the beam elements based onthe finite element (FE) method [15 16] The mass matrix of asingle member of the tower can be expressed as

M(119894) = T(119894)119879119890 M(119894)119890 T(119894)

119890(1)

in whichM(119894) andM(119894)119890 denote the element mass matrix of the119894th element in the global coordinate system (GCS) and thelocal coordinate system (LCS) respectively T(119894)119890 denotes thecoordinate transformation matrix of the 119894th element

Similarly the element stiffness matrix of the 119894th elementin the GCS K(119894) can be expressed as the multiple of theelement stiffness matrix K(119894)119890 in the LCS with the coordinatetransformation matrix T(119894)119890

K(119894) = T(119894)119879119890 K(119894)119890 T(119894)

119890 (2)

The global mass matrix M119905 and the stiffness matrix K119905 of atransmission tower can be expressed as follows

M119905 =119899119890

sum

119894=1

T(119894)119879M(119894)T(119894)

K119905 =119899119890

sum

119894=1

T(119894)119879K(119894)T(119894)

(3)

where 119899119890 is the number of elements of the tower model T(119894)is the freedom transform matrix from the LCS to the GCS

Shock and Vibration 3

v

1205791

1205792 12057931205794

1205795

l1l2 l3

l4

l5

u

(a) In-plane vibration

l2

l1

m1

m2

(b) Out-of-planevibration

Figure 2 MDOF elastic model of a transmission line

The three-dimensional (3D) FE model of a transmissiontower is displayed in Figure 1(a) If the configuration of atransmission tower is quite complicated due to a great num-ber of structural members and components a remarkablenumber of the degree of freedoms (DOFs) of the tower canbe expected Therefore it is impractical for the dynamicresponse computation and the investigation on the controlperformance because the numerical step-by-step integrationin the time domain will be very time-consuming In additionthe dynamic excitation on the tower such as wind loadscan commonly be modeled as a stationary or nonstationarystochastic process in time and nonhomogeneous in spaceThe simulation of dynamic loading of a complicate three-dimensional FE model of a transmission tower requiresenormous computation efforts From the practical viewpointa simple analytical model of the transmission tower withlimited DOFs is more reasonable for the investigation onstructural dynamic responses and the parametric study Inthis regard a two-dimensional (2D) lumped mass modelis commonly used in practice to examine the dynamicresponses of a complicated transmission tower

When a 3D FE dynamic model of a transmission toweris reduced to a 2D lumped mass model some assumptionsare commonly adopted As far as the 2D lumped model isconcerned the masses of the transmission tower includingthe masses of all structural components and all nonstructuralcomponents are first concentrated at several floors only Theaverage of the displacements of all nodes at a certain flooris defined as the nominal displacement of that floor in thatdirection Then the number of dynamic degrees of freedomof the 2D lumped mass model is the number of the selectedfloors The mass matrixM of the 2D lumped mass model is adiagonal matrix which can be expressed as

M =

[

[

[

[

[

[

[

[

1198981

1198982

d119898119895

d119898119899119898

]

]

]

]

]

]

]

]

(119895 = 1 2 119899119898) (4)

in which119898119895 denotes the mass of the 119895th floor 119899119898 is the floornumber of the transmission tower The stiffness matrix K ofthe 2D lumped mass model can be obtained based on the 3D

FE model of the transmission tower by taking the followingsteps

(1) Apply the same horizontal force at each node at the 119894thfloor of the 3D model such that the sum of all forcesequals 1

(2) Determine the horizontal displacement of each nodeat the 119895th floor Define the nominal displacementof the 119895th floor to have the flexibility coefficient120575119895119894 (119894 119895 = 1 2 119899119898)

(3) Form the flexibility matrixΨ and inverse the flexibil-ity matrix to obtain the 2D stiffness matrix K of thetransmission tower

22 Model of Transmission Tower-Line System The transmis-sion line can bemodeled as several lumpedmasses connectedwith many elastic elements [17 18] The analytical models ofa transmission line for in-planeout-of-plane vibration withsix lumped masses and five elastic elements are displayedin Figure 2 Regarding the in-plane vibration of the trans-mission line the kinetic energy and the potential energy ofa transmission line can be computed by using the general-ized velocity and generalized displacement respectively Theapplication of the Hamiltonrsquos variational principle can lead tothe equation ofmotion of a transmission line [19]The kineticenergy of the transmission line can be expressed based on theLagrange formulation

119879line =4

sum

119894=1

1

2

119898119894 (2

119894 + V2119894 )

= 119879line (

1205852

1205853

1205854 1205751198971 120575

1198972 120575

1198973 120575

1198974 120575

1198975)

(5)

where 120585 and 120575 are the generalized coordinates of a transmis-sion line related to the difference of the angle 120579 and length 119897respectively 119906119894 and V119894 are the displacement of the 119894th lumpedmass in the horizontal and vertical direction respectively

Similarly the potential energy of the transmission line isgiven by

119880line =4

sum

119894=1

119898119894119892V119894 +5

sum

119895=1

119864119860

2

(

(119897119895119904 + 120575119897119895)2

1198971198950

minus

119897

2119895119904

1198971198950

) (6)

4 Shock and Vibration

where119864 and119860 are Youngrsquosmodulus and the cross section areaof a transmission line 1198971198950 is the length of the 119895th element119897119895119904 is the length of the 119895th element after deformation Themass matrixMin

119897 can be determined by computing the partialdifferential of the kinetic energy to the generalized velocitySimilarly the stiffness matrix Kin

119897 can be determined bycomputing the partial differential of the potential energy tothe generalized displacement

Figure 2(b) displays the analytical model of transmissionline for the out-of-plane vibration The transmission linecan be taken as a hanging line with several lumped massesThe mass matrix Mout

119897 and stiffness matrix Kout119897 of the

transmission line can be expressed as

Mout119897 = [

1198981

1198982

]

Kout119897 =

[

[

[

[

1198981119892

1198971

minus

1198981119892

1198971

minus

1198981119892

1198971

1198981119892

1198971

+

(1198981 + 1198982) 119892

1198972

]

]

]

]

(7)

The kinetic energy of the tower-line system for the in-planevibration is given by

119879 =

119899119897

sum

119894=1

119879

119894

line +119899119905

sum

119895=1

119879

119895

tower (8)

in which nl and nt are the number of the transmissionlines and towers of the transmission tower-line systemrespectively Similarly the potential energy of the tower-linesystem is given by

119880 =

119899119897

sum

119894=1

119880

119894

line +119899119905

sum

119895=1

119880

119895

tower (9)

By substituting (8) and (9) into the Lagrange equation themass matrix Min and stiffness matrix Kin of a transmissiontower-line system for the in-plane vibration can be deter-mined by computing the partial differential of the kineticenergy 119879 and potential energy 119880 to generalize coordinatesand its first time derivatives Similarly the mass matrixMout and stiffness matrix Kout of the transmission tower-linesystem for the out-of-plane vibration can be determined byconsideration of the coupled effects induced by transmissionlines [1]

3 Model of Friction Damper

Passive friction dampers are the devices utilizing the mech-anism of the solid friction to achieve the energy dissipa-tion In reality this concept has been successfully used bymechanical engineers to control themotion ofmachinery andautomobiles The passive friction damper can be designedwith a specific optimum slip force under a particular externalexcitation (Figure 6) The development of friction damp-ing devices was pioneered in the early 1980s thereafter aconsiderable progress in developing a number of energy-dissipating devices in the structural vibration control [20ndash23] The configuration of a typical passive friction damper

Coefficient of friction 120583

Clamping force NSlippage e

d(t) d(t)

kd =EA

L

Control force fd(t)

Figure 3 Configuration of passive friction damper

fdfd

S e

k = EAL

Figure 4 Mechanical model of friction damper

is displayed in Figure 3 In the early stage the control forceof the friction damper is expressed as the product of theclamping force 119873 and the coefficient of friction 120583 withoutconsidering the effects of the damper axial stiffness 119896119889However it is found that the performance of the frictiondamper is tightly related to the axial stiffness of the damperand many mechanical models with the elastic axial memberare developed based on the experimental observations

Commonly the control force of a friction damper istransferred to the structure through an elastic axial memberThus the model of a friction damper should include thestiffness of the elastic member as shown in Figure 4 InFigure 4 119878 denotes the slip force of the 119894th friction damperwhich equals the product of the clamping force 119873 and thefriction coefficient 120583 119890 denotes the slip deformation of the119894th friction damper and it is equal to zero initially 119864 denotesYoungrsquos modulus of the damper119860 is the cross section area ofthe 119894th elastic member and 119871 is the length of the 119894th memberThe control force of the 119894th friction damper can be expressedas

119906119894 = 119906 (119896 119890 119878) (10)

where 119896 denotes the axial stiffness of a passive frictiondamper

The clamping force of a passive friction damper is set inadvance and its friction force is constant during the slippingprocess The condition and control force of the 119894th passivefriction damper can be determined using the following rules

Shock and Vibration 5

Yes

Yes

Yes

Yes

No

No

No

No

Stick stage

Stick stage

Slip stage

End

dt Δd e

dt+Δt = dt

t

+ Δdt

fdt+Δt = (dt+Δt minus e)EAL

fdt+ΔtΔdt gt 0 fdt+ΔtΔdt gt 0

e = e + Δdt

|fdt+Δt| = S

|fdt+Δt| = S

|fdt+Δt| gt S

e = e + (fdt+Δt minus S)LEA

Figure 5 Force flow chart of passive damper

(1) If the friction damper has no slippage then

119891

119894

119889 = (119889119894 minus 119890119894)119864119860 119894

119871 119894

119890119894 = 0

(11)

(2) If the friction damper has a slippage10038161003816100381610038161003816

119891

119894

119889

10038161003816100381610038161003816

= 119878119894 = 119891

119894

119889119873119894 sgn ( 119889119894)

119890119894 =

119889119894

(12)

where 119889119894 and 119889119894 are relative displacement and velocity

between the two ends of the 119894th friction damperrespectively Furthermore if the relative displacementincrement of the two ends of the 119894th friction elementΔ119889119894 is larger than zero the control force is tensileOtherwise the control force is compressive

(3) If the control force 119906119894 is of different sign with therelative displacement increment Δ119889119894 the frictiondamper is in the stick stage Under this circumstancethe resistant force and slippage of friction dampershould be determined following (11)

The flow chart for judging the slipping condition andthe control force of a friction damper is indicated inFigure 5

4 Equation of Motion of TransmissionTower-Line System

Thewind loading acting on a transmission tower-line systemcan be simulated with the aids of the spectral represen-tation approach by assuming the wind excitations to bea stationary stochastic process The stochastic wind fields

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

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Active and Passive Electronic Components

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

overcome the shortcoming of the dynamic absorbers manyenergy-dissipating dampers have been developed recentlyas an alternative approach for dynamic mitigation of trans-mission tower-line system Up to now the vibration controlof transmission tower-line system subjected to dynamicexcitations has been investigated rarely [11 12] Chen et alproposed a new method for the wind-resistant design of thetransmission tower-line system by using viscoelastic dampers[13] Chen et al also proposed a novel approach for the wind-induced semiactive vibration control of transmission tower-line system by using magnetorheological (MR) dampersHowever it is widely reported that the performance of theviscoelastic dampers can be substantially weakened underthe harsh environment which cannot be avoided for thetransmission tower-line system in the open air [10 14]The configuration and fabrication of the semiactive controldevices such as MR damper are quite complicated andthe requirement in the additional energy supply duringthe vibration control process is unrealistic while acceptingstrong excitations Therefore the passive friction damperswith a simple configuration and excellent environmentaladaptability can be adopted to suppress the wind-inducedresponses of the transmission tower-line systemThe frictiondamper is a typical energy-dissipating device utilized in thestructural response control application and it can be manu-factured as an axial member to replace common structuralmembers of a tower without requiring the additional spaceoccupancy

To this end the vibration control and performanceevaluation on a transmission tower-line system with passivefriction dampers subjected to wind loading is actively carriedout in this studyThe three-dimensional finite element modelof a transmission tower is firstly constructed and then a two-dimensional lumped mass model is also developed The ana-lytical model of transmission tower-line system is proposedby taking the dynamic interaction between the tower andthe transmission lines into consideration The mechanicalmodel of the friction damper is presented by consideringthe effects of damper axial stiffness The equation of motionof the transmission tower-line system incorporated with thefriction dampers disturbed by wind excitations is establishedfor both the in-plane and out-of-plane vibration respec-tively A real transmission tower-line system constructedin southern China is taken as an example to examine thefeasibility and reliability of the proposed control approach Inaddition a parametric study is conducted in detail in orderto examine the effects of the slipping force the axial stiffnessand the hysteresis loop on the control performanceThemadeobservations indicate that the passive friction dampers can beutilized to suppress the wind-induced vibration control of thetransmission tower-line system due to its satisfactory energy-dissipating capacity if the damper parameters are optimallydetermined

2 Model of Transmission Tower-Line System

21 Model of Transmission Tower To be a typical truss towerconstructed by using steelmembers a transmission tower can

9000

9800

7500

8000

107800

17000

12500

12500

98000

15000

13800

11200

12500

13000

(a) Three-dimensional FE model

107800

13000

25500

36700

50500

65500

78000

90500

980009

8

7

6

5

4

3

2

1

(b) Lumped massmodel

Figure 1 Analytical model of a transmission tower

be commonlymodelled by using the beam elements based onthe finite element (FE) method [15 16] The mass matrix of asingle member of the tower can be expressed as

M(119894) = T(119894)119879119890 M(119894)119890 T(119894)

119890(1)

in whichM(119894) andM(119894)119890 denote the element mass matrix of the119894th element in the global coordinate system (GCS) and thelocal coordinate system (LCS) respectively T(119894)119890 denotes thecoordinate transformation matrix of the 119894th element

Similarly the element stiffness matrix of the 119894th elementin the GCS K(119894) can be expressed as the multiple of theelement stiffness matrix K(119894)119890 in the LCS with the coordinatetransformation matrix T(119894)119890

K(119894) = T(119894)119879119890 K(119894)119890 T(119894)

119890 (2)

The global mass matrix M119905 and the stiffness matrix K119905 of atransmission tower can be expressed as follows

M119905 =119899119890

sum

119894=1

T(119894)119879M(119894)T(119894)

K119905 =119899119890

sum

119894=1

T(119894)119879K(119894)T(119894)

(3)

where 119899119890 is the number of elements of the tower model T(119894)is the freedom transform matrix from the LCS to the GCS

Shock and Vibration 3

v

1205791

1205792 12057931205794

1205795

l1l2 l3

l4

l5

u

(a) In-plane vibration

l2

l1

m1

m2

(b) Out-of-planevibration

Figure 2 MDOF elastic model of a transmission line

The three-dimensional (3D) FE model of a transmissiontower is displayed in Figure 1(a) If the configuration of atransmission tower is quite complicated due to a great num-ber of structural members and components a remarkablenumber of the degree of freedoms (DOFs) of the tower canbe expected Therefore it is impractical for the dynamicresponse computation and the investigation on the controlperformance because the numerical step-by-step integrationin the time domain will be very time-consuming In additionthe dynamic excitation on the tower such as wind loadscan commonly be modeled as a stationary or nonstationarystochastic process in time and nonhomogeneous in spaceThe simulation of dynamic loading of a complicate three-dimensional FE model of a transmission tower requiresenormous computation efforts From the practical viewpointa simple analytical model of the transmission tower withlimited DOFs is more reasonable for the investigation onstructural dynamic responses and the parametric study Inthis regard a two-dimensional (2D) lumped mass modelis commonly used in practice to examine the dynamicresponses of a complicated transmission tower

When a 3D FE dynamic model of a transmission toweris reduced to a 2D lumped mass model some assumptionsare commonly adopted As far as the 2D lumped model isconcerned the masses of the transmission tower includingthe masses of all structural components and all nonstructuralcomponents are first concentrated at several floors only Theaverage of the displacements of all nodes at a certain flooris defined as the nominal displacement of that floor in thatdirection Then the number of dynamic degrees of freedomof the 2D lumped mass model is the number of the selectedfloors The mass matrixM of the 2D lumped mass model is adiagonal matrix which can be expressed as

M =

[

[

[

[

[

[

[

[

1198981

1198982

d119898119895

d119898119899119898

]

]

]

]

]

]

]

]

(119895 = 1 2 119899119898) (4)

in which119898119895 denotes the mass of the 119895th floor 119899119898 is the floornumber of the transmission tower The stiffness matrix K ofthe 2D lumped mass model can be obtained based on the 3D

FE model of the transmission tower by taking the followingsteps

(1) Apply the same horizontal force at each node at the 119894thfloor of the 3D model such that the sum of all forcesequals 1

(2) Determine the horizontal displacement of each nodeat the 119895th floor Define the nominal displacementof the 119895th floor to have the flexibility coefficient120575119895119894 (119894 119895 = 1 2 119899119898)

(3) Form the flexibility matrixΨ and inverse the flexibil-ity matrix to obtain the 2D stiffness matrix K of thetransmission tower

22 Model of Transmission Tower-Line System The transmis-sion line can bemodeled as several lumpedmasses connectedwith many elastic elements [17 18] The analytical models ofa transmission line for in-planeout-of-plane vibration withsix lumped masses and five elastic elements are displayedin Figure 2 Regarding the in-plane vibration of the trans-mission line the kinetic energy and the potential energy ofa transmission line can be computed by using the general-ized velocity and generalized displacement respectively Theapplication of the Hamiltonrsquos variational principle can lead tothe equation ofmotion of a transmission line [19]The kineticenergy of the transmission line can be expressed based on theLagrange formulation

119879line =4

sum

119894=1

1

2

119898119894 (2

119894 + V2119894 )

= 119879line (

1205852

1205853

1205854 1205751198971 120575

1198972 120575

1198973 120575

1198974 120575

1198975)

(5)

where 120585 and 120575 are the generalized coordinates of a transmis-sion line related to the difference of the angle 120579 and length 119897respectively 119906119894 and V119894 are the displacement of the 119894th lumpedmass in the horizontal and vertical direction respectively

Similarly the potential energy of the transmission line isgiven by

119880line =4

sum

119894=1

119898119894119892V119894 +5

sum

119895=1

119864119860

2

(

(119897119895119904 + 120575119897119895)2

1198971198950

minus

119897

2119895119904

1198971198950

) (6)

4 Shock and Vibration

where119864 and119860 are Youngrsquosmodulus and the cross section areaof a transmission line 1198971198950 is the length of the 119895th element119897119895119904 is the length of the 119895th element after deformation Themass matrixMin

119897 can be determined by computing the partialdifferential of the kinetic energy to the generalized velocitySimilarly the stiffness matrix Kin

119897 can be determined bycomputing the partial differential of the potential energy tothe generalized displacement

Figure 2(b) displays the analytical model of transmissionline for the out-of-plane vibration The transmission linecan be taken as a hanging line with several lumped massesThe mass matrix Mout

119897 and stiffness matrix Kout119897 of the

transmission line can be expressed as

Mout119897 = [

1198981

1198982

]

Kout119897 =

[

[

[

[

1198981119892

1198971

minus

1198981119892

1198971

minus

1198981119892

1198971

1198981119892

1198971

+

(1198981 + 1198982) 119892

1198972

]

]

]

]

(7)

The kinetic energy of the tower-line system for the in-planevibration is given by

119879 =

119899119897

sum

119894=1

119879

119894

line +119899119905

sum

119895=1

119879

119895

tower (8)

in which nl and nt are the number of the transmissionlines and towers of the transmission tower-line systemrespectively Similarly the potential energy of the tower-linesystem is given by

119880 =

119899119897

sum

119894=1

119880

119894

line +119899119905

sum

119895=1

119880

119895

tower (9)

By substituting (8) and (9) into the Lagrange equation themass matrix Min and stiffness matrix Kin of a transmissiontower-line system for the in-plane vibration can be deter-mined by computing the partial differential of the kineticenergy 119879 and potential energy 119880 to generalize coordinatesand its first time derivatives Similarly the mass matrixMout and stiffness matrix Kout of the transmission tower-linesystem for the out-of-plane vibration can be determined byconsideration of the coupled effects induced by transmissionlines [1]

3 Model of Friction Damper

Passive friction dampers are the devices utilizing the mech-anism of the solid friction to achieve the energy dissipa-tion In reality this concept has been successfully used bymechanical engineers to control themotion ofmachinery andautomobiles The passive friction damper can be designedwith a specific optimum slip force under a particular externalexcitation (Figure 6) The development of friction damp-ing devices was pioneered in the early 1980s thereafter aconsiderable progress in developing a number of energy-dissipating devices in the structural vibration control [20ndash23] The configuration of a typical passive friction damper

Coefficient of friction 120583

Clamping force NSlippage e

d(t) d(t)

kd =EA

L

Control force fd(t)

Figure 3 Configuration of passive friction damper

fdfd

S e

k = EAL

Figure 4 Mechanical model of friction damper

is displayed in Figure 3 In the early stage the control forceof the friction damper is expressed as the product of theclamping force 119873 and the coefficient of friction 120583 withoutconsidering the effects of the damper axial stiffness 119896119889However it is found that the performance of the frictiondamper is tightly related to the axial stiffness of the damperand many mechanical models with the elastic axial memberare developed based on the experimental observations

Commonly the control force of a friction damper istransferred to the structure through an elastic axial memberThus the model of a friction damper should include thestiffness of the elastic member as shown in Figure 4 InFigure 4 119878 denotes the slip force of the 119894th friction damperwhich equals the product of the clamping force 119873 and thefriction coefficient 120583 119890 denotes the slip deformation of the119894th friction damper and it is equal to zero initially 119864 denotesYoungrsquos modulus of the damper119860 is the cross section area ofthe 119894th elastic member and 119871 is the length of the 119894th memberThe control force of the 119894th friction damper can be expressedas

119906119894 = 119906 (119896 119890 119878) (10)

where 119896 denotes the axial stiffness of a passive frictiondamper

The clamping force of a passive friction damper is set inadvance and its friction force is constant during the slippingprocess The condition and control force of the 119894th passivefriction damper can be determined using the following rules

Shock and Vibration 5

Yes

Yes

Yes

Yes

No

No

No

No

Stick stage

Stick stage

Slip stage

End

dt Δd e

dt+Δt = dt

t

+ Δdt

fdt+Δt = (dt+Δt minus e)EAL

fdt+ΔtΔdt gt 0 fdt+ΔtΔdt gt 0

e = e + Δdt

|fdt+Δt| = S

|fdt+Δt| = S

|fdt+Δt| gt S

e = e + (fdt+Δt minus S)LEA

Figure 5 Force flow chart of passive damper

(1) If the friction damper has no slippage then

119891

119894

119889 = (119889119894 minus 119890119894)119864119860 119894

119871 119894

119890119894 = 0

(11)

(2) If the friction damper has a slippage10038161003816100381610038161003816

119891

119894

119889

10038161003816100381610038161003816

= 119878119894 = 119891

119894

119889119873119894 sgn ( 119889119894)

119890119894 =

119889119894

(12)

where 119889119894 and 119889119894 are relative displacement and velocity

between the two ends of the 119894th friction damperrespectively Furthermore if the relative displacementincrement of the two ends of the 119894th friction elementΔ119889119894 is larger than zero the control force is tensileOtherwise the control force is compressive

(3) If the control force 119906119894 is of different sign with therelative displacement increment Δ119889119894 the frictiondamper is in the stick stage Under this circumstancethe resistant force and slippage of friction dampershould be determined following (11)

The flow chart for judging the slipping condition andthe control force of a friction damper is indicated inFigure 5

4 Equation of Motion of TransmissionTower-Line System

Thewind loading acting on a transmission tower-line systemcan be simulated with the aids of the spectral represen-tation approach by assuming the wind excitations to bea stationary stochastic process The stochastic wind fields

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

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Shock and Vibration

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

v

1205791

1205792 12057931205794

1205795

l1l2 l3

l4

l5

u

(a) In-plane vibration

l2

l1

m1

m2

(b) Out-of-planevibration

Figure 2 MDOF elastic model of a transmission line

The three-dimensional (3D) FE model of a transmissiontower is displayed in Figure 1(a) If the configuration of atransmission tower is quite complicated due to a great num-ber of structural members and components a remarkablenumber of the degree of freedoms (DOFs) of the tower canbe expected Therefore it is impractical for the dynamicresponse computation and the investigation on the controlperformance because the numerical step-by-step integrationin the time domain will be very time-consuming In additionthe dynamic excitation on the tower such as wind loadscan commonly be modeled as a stationary or nonstationarystochastic process in time and nonhomogeneous in spaceThe simulation of dynamic loading of a complicate three-dimensional FE model of a transmission tower requiresenormous computation efforts From the practical viewpointa simple analytical model of the transmission tower withlimited DOFs is more reasonable for the investigation onstructural dynamic responses and the parametric study Inthis regard a two-dimensional (2D) lumped mass modelis commonly used in practice to examine the dynamicresponses of a complicated transmission tower

When a 3D FE dynamic model of a transmission toweris reduced to a 2D lumped mass model some assumptionsare commonly adopted As far as the 2D lumped model isconcerned the masses of the transmission tower includingthe masses of all structural components and all nonstructuralcomponents are first concentrated at several floors only Theaverage of the displacements of all nodes at a certain flooris defined as the nominal displacement of that floor in thatdirection Then the number of dynamic degrees of freedomof the 2D lumped mass model is the number of the selectedfloors The mass matrixM of the 2D lumped mass model is adiagonal matrix which can be expressed as

M =

[

[

[

[

[

[

[

[

1198981

1198982

d119898119895

d119898119899119898

]

]

]

]

]

]

]

]

(119895 = 1 2 119899119898) (4)

in which119898119895 denotes the mass of the 119895th floor 119899119898 is the floornumber of the transmission tower The stiffness matrix K ofthe 2D lumped mass model can be obtained based on the 3D

FE model of the transmission tower by taking the followingsteps

(1) Apply the same horizontal force at each node at the 119894thfloor of the 3D model such that the sum of all forcesequals 1

(2) Determine the horizontal displacement of each nodeat the 119895th floor Define the nominal displacementof the 119895th floor to have the flexibility coefficient120575119895119894 (119894 119895 = 1 2 119899119898)

(3) Form the flexibility matrixΨ and inverse the flexibil-ity matrix to obtain the 2D stiffness matrix K of thetransmission tower

22 Model of Transmission Tower-Line System The transmis-sion line can bemodeled as several lumpedmasses connectedwith many elastic elements [17 18] The analytical models ofa transmission line for in-planeout-of-plane vibration withsix lumped masses and five elastic elements are displayedin Figure 2 Regarding the in-plane vibration of the trans-mission line the kinetic energy and the potential energy ofa transmission line can be computed by using the general-ized velocity and generalized displacement respectively Theapplication of the Hamiltonrsquos variational principle can lead tothe equation ofmotion of a transmission line [19]The kineticenergy of the transmission line can be expressed based on theLagrange formulation

119879line =4

sum

119894=1

1

2

119898119894 (2

119894 + V2119894 )

= 119879line (

1205852

1205853

1205854 1205751198971 120575

1198972 120575

1198973 120575

1198974 120575

1198975)

(5)

where 120585 and 120575 are the generalized coordinates of a transmis-sion line related to the difference of the angle 120579 and length 119897respectively 119906119894 and V119894 are the displacement of the 119894th lumpedmass in the horizontal and vertical direction respectively

Similarly the potential energy of the transmission line isgiven by

119880line =4

sum

119894=1

119898119894119892V119894 +5

sum

119895=1

119864119860

2

(

(119897119895119904 + 120575119897119895)2

1198971198950

minus

119897

2119895119904

1198971198950

) (6)

4 Shock and Vibration

where119864 and119860 are Youngrsquosmodulus and the cross section areaof a transmission line 1198971198950 is the length of the 119895th element119897119895119904 is the length of the 119895th element after deformation Themass matrixMin

119897 can be determined by computing the partialdifferential of the kinetic energy to the generalized velocitySimilarly the stiffness matrix Kin

119897 can be determined bycomputing the partial differential of the potential energy tothe generalized displacement

Figure 2(b) displays the analytical model of transmissionline for the out-of-plane vibration The transmission linecan be taken as a hanging line with several lumped massesThe mass matrix Mout

119897 and stiffness matrix Kout119897 of the

transmission line can be expressed as

Mout119897 = [

1198981

1198982

]

Kout119897 =

[

[

[

[

1198981119892

1198971

minus

1198981119892

1198971

minus

1198981119892

1198971

1198981119892

1198971

+

(1198981 + 1198982) 119892

1198972

]

]

]

]

(7)

The kinetic energy of the tower-line system for the in-planevibration is given by

119879 =

119899119897

sum

119894=1

119879

119894

line +119899119905

sum

119895=1

119879

119895

tower (8)

in which nl and nt are the number of the transmissionlines and towers of the transmission tower-line systemrespectively Similarly the potential energy of the tower-linesystem is given by

119880 =

119899119897

sum

119894=1

119880

119894

line +119899119905

sum

119895=1

119880

119895

tower (9)

By substituting (8) and (9) into the Lagrange equation themass matrix Min and stiffness matrix Kin of a transmissiontower-line system for the in-plane vibration can be deter-mined by computing the partial differential of the kineticenergy 119879 and potential energy 119880 to generalize coordinatesand its first time derivatives Similarly the mass matrixMout and stiffness matrix Kout of the transmission tower-linesystem for the out-of-plane vibration can be determined byconsideration of the coupled effects induced by transmissionlines [1]

3 Model of Friction Damper

Passive friction dampers are the devices utilizing the mech-anism of the solid friction to achieve the energy dissipa-tion In reality this concept has been successfully used bymechanical engineers to control themotion ofmachinery andautomobiles The passive friction damper can be designedwith a specific optimum slip force under a particular externalexcitation (Figure 6) The development of friction damp-ing devices was pioneered in the early 1980s thereafter aconsiderable progress in developing a number of energy-dissipating devices in the structural vibration control [20ndash23] The configuration of a typical passive friction damper

Coefficient of friction 120583

Clamping force NSlippage e

d(t) d(t)

kd =EA

L

Control force fd(t)

Figure 3 Configuration of passive friction damper

fdfd

S e

k = EAL

Figure 4 Mechanical model of friction damper

is displayed in Figure 3 In the early stage the control forceof the friction damper is expressed as the product of theclamping force 119873 and the coefficient of friction 120583 withoutconsidering the effects of the damper axial stiffness 119896119889However it is found that the performance of the frictiondamper is tightly related to the axial stiffness of the damperand many mechanical models with the elastic axial memberare developed based on the experimental observations

Commonly the control force of a friction damper istransferred to the structure through an elastic axial memberThus the model of a friction damper should include thestiffness of the elastic member as shown in Figure 4 InFigure 4 119878 denotes the slip force of the 119894th friction damperwhich equals the product of the clamping force 119873 and thefriction coefficient 120583 119890 denotes the slip deformation of the119894th friction damper and it is equal to zero initially 119864 denotesYoungrsquos modulus of the damper119860 is the cross section area ofthe 119894th elastic member and 119871 is the length of the 119894th memberThe control force of the 119894th friction damper can be expressedas

119906119894 = 119906 (119896 119890 119878) (10)

where 119896 denotes the axial stiffness of a passive frictiondamper

The clamping force of a passive friction damper is set inadvance and its friction force is constant during the slippingprocess The condition and control force of the 119894th passivefriction damper can be determined using the following rules

Shock and Vibration 5

Yes

Yes

Yes

Yes

No

No

No

No

Stick stage

Stick stage

Slip stage

End

dt Δd e

dt+Δt = dt

t

+ Δdt

fdt+Δt = (dt+Δt minus e)EAL

fdt+ΔtΔdt gt 0 fdt+ΔtΔdt gt 0

e = e + Δdt

|fdt+Δt| = S

|fdt+Δt| = S

|fdt+Δt| gt S

e = e + (fdt+Δt minus S)LEA

Figure 5 Force flow chart of passive damper

(1) If the friction damper has no slippage then

119891

119894

119889 = (119889119894 minus 119890119894)119864119860 119894

119871 119894

119890119894 = 0

(11)

(2) If the friction damper has a slippage10038161003816100381610038161003816

119891

119894

119889

10038161003816100381610038161003816

= 119878119894 = 119891

119894

119889119873119894 sgn ( 119889119894)

119890119894 =

119889119894

(12)

where 119889119894 and 119889119894 are relative displacement and velocity

between the two ends of the 119894th friction damperrespectively Furthermore if the relative displacementincrement of the two ends of the 119894th friction elementΔ119889119894 is larger than zero the control force is tensileOtherwise the control force is compressive

(3) If the control force 119906119894 is of different sign with therelative displacement increment Δ119889119894 the frictiondamper is in the stick stage Under this circumstancethe resistant force and slippage of friction dampershould be determined following (11)

The flow chart for judging the slipping condition andthe control force of a friction damper is indicated inFigure 5

4 Equation of Motion of TransmissionTower-Line System

Thewind loading acting on a transmission tower-line systemcan be simulated with the aids of the spectral represen-tation approach by assuming the wind excitations to bea stationary stochastic process The stochastic wind fields

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

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4 Shock and Vibration

where119864 and119860 are Youngrsquosmodulus and the cross section areaof a transmission line 1198971198950 is the length of the 119895th element119897119895119904 is the length of the 119895th element after deformation Themass matrixMin

119897 can be determined by computing the partialdifferential of the kinetic energy to the generalized velocitySimilarly the stiffness matrix Kin

119897 can be determined bycomputing the partial differential of the potential energy tothe generalized displacement

Figure 2(b) displays the analytical model of transmissionline for the out-of-plane vibration The transmission linecan be taken as a hanging line with several lumped massesThe mass matrix Mout

119897 and stiffness matrix Kout119897 of the

transmission line can be expressed as

Mout119897 = [

1198981

1198982

]

Kout119897 =

[

[

[

[

1198981119892

1198971

minus

1198981119892

1198971

minus

1198981119892

1198971

1198981119892

1198971

+

(1198981 + 1198982) 119892

1198972

]

]

]

]

(7)

The kinetic energy of the tower-line system for the in-planevibration is given by

119879 =

119899119897

sum

119894=1

119879

119894

line +119899119905

sum

119895=1

119879

119895

tower (8)

in which nl and nt are the number of the transmissionlines and towers of the transmission tower-line systemrespectively Similarly the potential energy of the tower-linesystem is given by

119880 =

119899119897

sum

119894=1

119880

119894

line +119899119905

sum

119895=1

119880

119895

tower (9)

By substituting (8) and (9) into the Lagrange equation themass matrix Min and stiffness matrix Kin of a transmissiontower-line system for the in-plane vibration can be deter-mined by computing the partial differential of the kineticenergy 119879 and potential energy 119880 to generalize coordinatesand its first time derivatives Similarly the mass matrixMout and stiffness matrix Kout of the transmission tower-linesystem for the out-of-plane vibration can be determined byconsideration of the coupled effects induced by transmissionlines [1]

3 Model of Friction Damper

Passive friction dampers are the devices utilizing the mech-anism of the solid friction to achieve the energy dissipa-tion In reality this concept has been successfully used bymechanical engineers to control themotion ofmachinery andautomobiles The passive friction damper can be designedwith a specific optimum slip force under a particular externalexcitation (Figure 6) The development of friction damp-ing devices was pioneered in the early 1980s thereafter aconsiderable progress in developing a number of energy-dissipating devices in the structural vibration control [20ndash23] The configuration of a typical passive friction damper

Coefficient of friction 120583

Clamping force NSlippage e

d(t) d(t)

kd =EA

L

Control force fd(t)

Figure 3 Configuration of passive friction damper

fdfd

S e

k = EAL

Figure 4 Mechanical model of friction damper

is displayed in Figure 3 In the early stage the control forceof the friction damper is expressed as the product of theclamping force 119873 and the coefficient of friction 120583 withoutconsidering the effects of the damper axial stiffness 119896119889However it is found that the performance of the frictiondamper is tightly related to the axial stiffness of the damperand many mechanical models with the elastic axial memberare developed based on the experimental observations

Commonly the control force of a friction damper istransferred to the structure through an elastic axial memberThus the model of a friction damper should include thestiffness of the elastic member as shown in Figure 4 InFigure 4 119878 denotes the slip force of the 119894th friction damperwhich equals the product of the clamping force 119873 and thefriction coefficient 120583 119890 denotes the slip deformation of the119894th friction damper and it is equal to zero initially 119864 denotesYoungrsquos modulus of the damper119860 is the cross section area ofthe 119894th elastic member and 119871 is the length of the 119894th memberThe control force of the 119894th friction damper can be expressedas

119906119894 = 119906 (119896 119890 119878) (10)

where 119896 denotes the axial stiffness of a passive frictiondamper

The clamping force of a passive friction damper is set inadvance and its friction force is constant during the slippingprocess The condition and control force of the 119894th passivefriction damper can be determined using the following rules

Shock and Vibration 5

Yes

Yes

Yes

Yes

No

No

No

No

Stick stage

Stick stage

Slip stage

End

dt Δd e

dt+Δt = dt

t

+ Δdt

fdt+Δt = (dt+Δt minus e)EAL

fdt+ΔtΔdt gt 0 fdt+ΔtΔdt gt 0

e = e + Δdt

|fdt+Δt| = S

|fdt+Δt| = S

|fdt+Δt| gt S

e = e + (fdt+Δt minus S)LEA

Figure 5 Force flow chart of passive damper

(1) If the friction damper has no slippage then

119891

119894

119889 = (119889119894 minus 119890119894)119864119860 119894

119871 119894

119890119894 = 0

(11)

(2) If the friction damper has a slippage10038161003816100381610038161003816

119891

119894

119889

10038161003816100381610038161003816

= 119878119894 = 119891

119894

119889119873119894 sgn ( 119889119894)

119890119894 =

119889119894

(12)

where 119889119894 and 119889119894 are relative displacement and velocity

between the two ends of the 119894th friction damperrespectively Furthermore if the relative displacementincrement of the two ends of the 119894th friction elementΔ119889119894 is larger than zero the control force is tensileOtherwise the control force is compressive

(3) If the control force 119906119894 is of different sign with therelative displacement increment Δ119889119894 the frictiondamper is in the stick stage Under this circumstancethe resistant force and slippage of friction dampershould be determined following (11)

The flow chart for judging the slipping condition andthe control force of a friction damper is indicated inFigure 5

4 Equation of Motion of TransmissionTower-Line System

Thewind loading acting on a transmission tower-line systemcan be simulated with the aids of the spectral represen-tation approach by assuming the wind excitations to bea stationary stochastic process The stochastic wind fields

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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Acoustics and VibrationAdvances in

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Advances inOptoElectronics

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

Yes

Yes

Yes

Yes

No

No

No

No

Stick stage

Stick stage

Slip stage

End

dt Δd e

dt+Δt = dt

t

+ Δdt

fdt+Δt = (dt+Δt minus e)EAL

fdt+ΔtΔdt gt 0 fdt+ΔtΔdt gt 0

e = e + Δdt

|fdt+Δt| = S

|fdt+Δt| = S

|fdt+Δt| gt S

e = e + (fdt+Δt minus S)LEA

Figure 5 Force flow chart of passive damper

(1) If the friction damper has no slippage then

119891

119894

119889 = (119889119894 minus 119890119894)119864119860 119894

119871 119894

119890119894 = 0

(11)

(2) If the friction damper has a slippage10038161003816100381610038161003816

119891

119894

119889

10038161003816100381610038161003816

= 119878119894 = 119891

119894

119889119873119894 sgn ( 119889119894)

119890119894 =

119889119894

(12)

where 119889119894 and 119889119894 are relative displacement and velocity

between the two ends of the 119894th friction damperrespectively Furthermore if the relative displacementincrement of the two ends of the 119894th friction elementΔ119889119894 is larger than zero the control force is tensileOtherwise the control force is compressive

(3) If the control force 119906119894 is of different sign with therelative displacement increment Δ119889119894 the frictiondamper is in the stick stage Under this circumstancethe resistant force and slippage of friction dampershould be determined following (11)

The flow chart for judging the slipping condition andthe control force of a friction damper is indicated inFigure 5

4 Equation of Motion of TransmissionTower-Line System

Thewind loading acting on a transmission tower-line systemcan be simulated with the aids of the spectral represen-tation approach by assuming the wind excitations to bea stationary stochastic process The stochastic wind fields

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

9

8

7

6

5

4

3

2

1Joint

Friction damper

Figure 6 Installation scheme of friction damper

119901

0119895 (119905) 119895 = 1 2 119899119898 can be simulated by using the follow-

ing series [24]

119901119895 (119905)

= 2radicΔ120596

119895

sum

119898=1

119873

sum

119897=1

10038161003816100381610038161003816

119867119895119898 (120596119898119897)10038161003816100381610038161003816

cos (120596119898119897119905 minus 120579119895119898 (120596119898119897) + 120593119898119897)

Δ120596 =

120596up

119873

(13)

where 119873 is a sufficiently large number Δ120596 is the frequencyincrement 120596up is the upper cutoff frequency with the con-dition that when 120596 gt 120596up the value of the cross spectraldensity matrix S0(120596) is trivial 120601119898119897 represents a sequenceof independent random phase angles uniformly distributedover the interval [0 2120587]119867119895119898(120596) is a typical element ofmatrixH(120596) which is defined with the Cholesky decomposition ofthe S0(120596) 120579119895119898(120596) is the complex angle of119867119895119898(120596)

The equation of motion of the transmission tower-linesystem for in-plane vibration subjected to wind loading isgiven by

Minxin (119905) + Cinxin (119905) + Kinxin (119905) = Pin(119905) (14)

where xin(119905) xin(119905) and xin(119905) are the displacement velocityand acceleration responses for the in-plane vibration respec-tively Min Cin and Kin are mass and damping and stiffnessmatrices of the tower-line system for the in-plane vibrationrespectively Pin

(119905) is the wind loading vector for the in-plane

Table 1 Parameters of friction damper

Youngrsquos modulus(Nm)

Cross section area(m2)

Maximum slippage(m)

206 times 10

1175 times 10

minus4 008

vibration Similarly the equation of motion for the out-of-plane vibration can be expressed as

Moutxout (119905) + Coutxout (119905) + Koutxout (119905) = Pout(119905) (15)

The meanings of the symbols in (15) are similar to those in(14)

The equations of motion of the transmission tower-linesystem with friction dampers for both the in-plane and out-of-plane vibration can be expressed as follows

Minxin (119905) + Cinxin (119905) + Kxin (119905) = Pin(119905) +Hinf in119889 (119905)

Moutxout (119905) + Coutxout (119905) + Kxout (119905)

= Pout(119905) +Houtfout119889 (119905)

(16)

where f in119889 (119905) and fout119889 (119905) are the control forces of frictiondampers for in-plane vibration and out-of-plane vibrationrespectively Hin and Hout are the influence matrix for thecontrol forces for the in-plane vibration and out-of-planevibration respectively

5 Case Study

To examine the feasibility of the proposed control approachbased on friction dampers a real transmission tower-linesystem constructed inChina is taken as the example structureto investigate the control performanceThe tower has a heightof 1078m and the span of the transmission line is 400mas shown in Figure 1 The axial rigidity EA of the transmis-sion line is 48 times 107N and the weight per meter of thetransmission line is 1394 kNm A 3D model is constructedbased on the FEmethod and a two-dimensional lumpedmassmodel is also developedThewind excitations are numericallysimulated by using the spectral representation method TheRayleigh damping assumption is adopted to construct thestructural damping matrix The damping ratios in the firsttwo modes of vibration of the tower are assumed to be 002Twenty passive friction dampers are evenly distributed alongthe main body of the tower with ten dampers placed in thein-plane direction and the other ten dampers placed in theout-of plane direction Table 1 lists the physical parametersof the passive friction damper adopted in this study for thewind-induced vibration mitigation

The comparison between the maximum responses with-outwith friction dampers is carried out for both the in-plane and out-of-plane vibration respectively Four scenariosof slipping forces of the friction dampers are selected forthe assessment on the control performance namely 20 kN40 kN 60 kN and 80 kN Displayed in Figure 7 are themaximum displacement velocity and acceleration responsesof the transmission tower under different slipping forces of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

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Active and Passive Electronic Components

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Navigation and Observation

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

0

1

2

3

4

5

6

7

8

9

00 02 04 06

In-plane

Displacement (m)

Mas

s

(a)

0

1

2

3

4

5

6

7

8

9

00 02 04

Out-of-plane

Displacement (m)

Mas

s

(b)

0

1

2

3

4

5

6

7

8

9

00 05 10 15 20

In-plane

Velocity (ms)

Mas

s

(c)

0

1

2

3

4

5

6

7

8

9

00 05 10 15

Out-of-plane

Velocity (ms)

Mas

s

(d)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8

In-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(e)

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Out-of-plane

Mas

s

Original towerfd = 20kN

fd = 40kN

fd = 60kNfd = 80kN

Acceleration (ms2)

(f)

Figure 7 Maximum responses under different damper force

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 20 40 60 80 10000

02

04

06 In-plane

Disp

lace

men

t (m

)

Control force (kN)

(a)

0 20 40 60 80 10000

02

04Out-of-plane

Disp

lace

men

t (m

)

Control force (kN)

(b)

0 20 40 60 80 10000

05

10

15

20In-plane

Velo

city

(ms

)

Control force (kN)

(c)

0 20 40 60 80 10000

05

10

15Out-of-plane

Velo

city

(ms

)

Control force (kN)

(d)

0 20 40 60 80 1000

2

4

6

8In-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

0 20 40 60 80 1000

2

4

6

8Out-of-plane

Control force (kN)

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 8 Variation of maximum responses with damper force

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

friction dampers in comparison with those without controlIt is seen that the peak dynamic responses of the trans-mission tower can be remarkably reduced by incorporatingthe friction dampers As far as the in-plane vibration isconcerned the peak displacement is gradually reduced withthe increasing damper slipping force The control scenariowith 80 kN presents the best control performance on thepeak displacement responses despite the peak velocity andacceleration responses Similar observations can be madefrom the results of the out-of-plane vibration Therefore theinstallation of the friction dampers can substantially suppressthe wind-induced vibration of the transmission tower-linesystem for both the in-plane and out-of-plane vibrationrespectively

6 Parametric Investigation

61 Effects of Damper Force The variations of peak responsesof tower with respect to the slipping forces of the frictiondampers are examined and displayed in Figure 8 Threetypical positions are selected namely the tower top (massnumber 9) the crossarm (mass number 8) and the top of thetower body (mass number 6) It is seen from Figure 8(a) thatthe peak displacement at the tower top crossarm and top oftower body gradually decreases when increasing the slippingforces of the friction dampers Similar variation trends can befound for the out-of-plane vibration while the peak displace-ment slightly increases when the slipping force is larger than20 kN The variations of peak velocity displayed in Figures8(b) and 8(c) present different characteristics compared tothe peak displacement The peak velocity and accelerationresponses dramatically reduce at the beginning when thedamper control forces are small Then the increment in thecontrol forces of the friction damper has almost no efficacy onthe peak velocity and acceleration responses for the in-planevibration For the out-of-plane vibration the continuousincrement in the control force may induce the slight increaseof the peak velocity and acceleration responses The sameobservations can be made from the dynamic responses of theother lumped masses such as the crossarm and the top of thetower body (mass number 6)

It is seen that no matter which tower mass is selectedthere is an optimal value of control forces by which thebest performance of the friction damper can be achievedTo compare the control performance of the three types ofthe dynamic responses of different places one can find thatthere exists a small difference for the displacement velocityand acceleration responses The optimal slipping forces ofthe friction dampers are about 40 kN and 20 kN for the in-plane vibration and out-of-plane vibration respectively If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyA possible explanation of such a phenomenon is that theunlimited increment in control forces may induce the stick offriction dampers when subjected to strong wind excitationsUnder these circumstances the friction dampers have noslippage and behave the same as a brace The displacementresponses may be reduced due to the stiffness incrementof the tower while the velocity and acceleration responses

cannot be improved because the damper has no slippageand energy dissipation Similar results can be observed forother lumped masses Therefore the optimal slipping forcesof the friction dampers are set as 40 kN and 20 kN for thein-plane vibration and out-of-plane vibration respectively inthe parametric investigation

62 Effects of Damper Stiffness Figure 9 displays the vari-ations of the peak displacement velocity and accelerationresponses of the tower with respect to the brace stiffness offriction dampers The stiffness coefficient of the damper isdefined as

SC =

119896

119889

119896

1198890

(17)

where 119896119889 denotes the damper stiffness used in the analysisand 1198961198890 denotes the stiffness parameter of a friction damper

It is seen that the all the three types of dynamic responsesfor both the in-plane and out-of-plane vibration reducerapidly when the stiffness coefficient of dampers increasesfrom 00 to about 10 Afterwards the peak responses varyslightlywith increasing stiffness coefficient of dampersWhenthe stiffness coefficient exceeds 20 the damper stiffness hasalmost no effects on the structural peak responses Similarresults can be observed for other lumped masses as shown inFigure 9 As a result the optimum stiffness coefficient of thefriction dampers for the wind-induced response control ofthe example transmission tower is about 10 Further increasein the damper stiffness cannot substantially improve thecontrol performance and leads to unnecessary cost increases

63 Comparison on Hysteresis Loops It is noted that theenclosed area of a hysteresis loop reflects the energy dis-sipated by the friction damper when subjected to dynamicexcitations Thus the hysteresis loop with large enclosedarea indicates the satisfactory control performance of thefriction dampers To this end the present section containsthe comparison on the force-displacement responses of thefriction damperswith different stiffness coefficients Figure 10displays the hysteresis loops of a friction damper for the in-plane vibrationThe slipping forces of all the friction dampersfor the in-plane vibration are set as 40 kN In reality theaxial stiffness of the friction damper can be reflected bythe slope of the hysteresis loop It is seen from Figure 10(a)that if the axial stiffness is small there will be a distinctaxial deformation of the damper which means that thedamper slippage is small and the energy-dissipating of thedamper is limited To increase the damper axial stiffnessthe friction damper is easy to slide and dissipate the vibrantenergy If the axial stiffness of a friction damper is largeenough under the unchanged slipping force its effects onthe hysteresis loops keep constant and thus the improvementon control performance is minor Displayed in Figure 11is the comparison on hysteresis loops of friction damperswith different slipping forces and the same axial stiffnessfor the in-plane vibration It is seen that the slope of thehysteresis loops of the damper decreases with the increasing

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

00 05 10 15 20 25 30 35 4000

02

04

06

08In-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(a)

00 05 10 15 20 25 30 35 4000

02

04Out-of-plane

Disp

lace

men

t (m

)

Coefficient of damper stiffness

(b)

00 05 10 15 20 25 30 35 4000

05

10

15

20In-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(c)

00 05 10 15 20 25 30 35 4000

05

10

15

20Out-of-plane

Velo

city

(ms

)

Coefficient of damper stiffness

(d)

00 05 10 15 20 25 30 35 400

2

4

6

8In-plane

Coefficient of damper stiffness

Mass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(e)

00 05 10 15 20 25 30 35 400

2

4

6

8Out-of-plane

Coefficient of damper stiffnessMass number 9 (top) Mass number 8 (crossarm) Mass number 6

Acce

lera

tion

(ms2)

(f)

Figure 9 Variation of maximum responses with damper stiffness

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

000 002 004

0

30

60In-plane

Con

trol f

orce

(kN

)

Slippage (m)

SC = 02

minus002minus004minus60

minus30

(a)

000 002 004

In-plane

Slippage (m)

= 06

minus002minus004

0

30

60

Con

trol f

orce

(kN

)

minus60

minus30

SC

(b)

0

000 001 002 003 004

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002 minus001

30

60

minus60

minus30

SC

(c)

Damper number 9

000 002 004

0

In-plane

Con

trol f

orce

(kN

)

Slippage (m)

= 20

minus004 minus002

30

60

minus60

minus30

SC

(d)

Figure 10 Hysteresis loops of friction damper for the in-plane vibration

000 002 004

0

50

100

150 In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus150

minus100

minus50

SC

(a)

000 002 004

0

10

20In-plane

Con

trol f

orce

(kN

)

Slippage (m)

Damper number 9

= 10

minus002minus004minus20

minus10

SC

(b)

Figure 11 Comparison on hysteresis loops of friction damper for the in-plane vibration

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

slipping forces Therefore the control performance cannotbe substantially improved by merely increasing the slippingforces of the damperThe axial stiffness of the friction dampershould be increased accordingly with the increasing damperslipping force Similar observations can be made from thecontrol evaluation for the out-of-plane vibration which is notdisplayed due to page limitation It can be concluded that theoptimal axial stiffness of the friction damper is tightly relatedto the damper slipping force A large value of the optimalaxial stiffness is expected if the slipping force of the damper islarge

7 Concluding Remarks

The feasibility of using passive friction dampers to suppressthe dynamic responses of a transmission tower-line systemunder wind excitations is performed in this study A 2Dlumped mass model of a transmission tower is developedfor the dynamic analysis by simplifying the 3D FE modelThe analytical model for the transmission tower-line systemis developed by simulating the transmission line as severallumpedmasses connectedwith elastic elementsThemechan-ical model of the passive friction damper is presented byinvolving the effects of the damper axial stiffness A realtransmission tower-line system is taken as the example toinvestigate the efficacy of the proposed control approachthrough the detailed parametric study

The made observations demonstrate that the installationof the friction dampers can substantially suppress the wind-induced vibration of the transmission tower-line system forboth the in-plane and out-of-plane vibration respectivelybecause of its satisfactory energy-dissipating capacity It isseen that no matter which tower mass is selected thereis an optimal value of control forces by which the bestperformance of the friction damper can be achieved If thedamper control force is increased after reaching the optimalvalue the damper performance will deteriorate graduallyThe dynamic responses for both the in-plane and out-of-plane vibration can be reduced rapidly when the stiffnesscoefficient of dampers increases fromzero to the optimal axialstiffness value Further increase in damper stiffness cannotsubstantially improve the control performance and leads tounnecessary cost increases The optimal axial stiffness of thefriction damper is tightly related to the damper slipping force

Conflict of Interests

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

Acknowledgments

The authors are grateful for the financial support from theNational Natural Science Foundation of China (51178366)the technological project of theChinese Southern PowerGridCo Ltd (K-GD2013-0783) the Fok Ying-Tong EducationFoundation (131072) and the Natural Science Foundation ofHubei Province (2014CFA026)

References

[1] B Chen J Zheng and W 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

[2] H-F Bai T-H Yi H-N Li and L Ren ldquoMultisensors on-sitemonitoring and characteristic analysis of UHV transmissiontowerrdquo International Journal of Distributed Sensor Networks vol2012 Article ID 545148 10 pages 2012

[3] E Simiu and R ScanlanWind Effects on Structures JohnWileyand Sons New York NY USA 3rd edition 1996

[4] Q Xie and L Sun ldquoFailure mechanism and retrofitting strategyof transmission tower structures under ice loadrdquo Journal ofConstructional Steel Research vol 74 pp 26ndash36 2012

[5] E Savory G A R Parke M Zeinoddini N Toy and PDisney ldquoModelling of tornado and microburst-induced windloading and failure of a lattice transmission towerrdquo EngineeringStructures vol 23 no 4 pp 365ndash375 2001

[6] 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

[7] H-N Li S-Y Tang and T-H Yi ldquoWind-rain-induced vibra-tion test and analytical method of high-voltage transmissiontowerrdquo Structural Engineering and Mechanics vol 48 no 4 pp435ndash453 2013

[8] P S Lee and G McClure ldquoElastoplastic large deformationanalysis of a lattice steel tower structure and comparison withfull-scale testsrdquo Journal of Constructional Steel Research vol 63no 5 pp 709ndash717 2007

[9] T Okamura T Ohkuma E Hongo and H Okada ldquoWindresponse analysis of a transmission tower in a mountainousareardquo Journal ofWind Engineering and Industrial Aerodynamicsvol 91 no 1-2 pp 53ndash63 2003

[10] 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

[11] S Ozono and J Maeda ldquoIn-plane dynamic interaction betweena tower and conductors at lower frequenciesrdquo EngineeringStructures vol 14 no 4 pp 210ndash216 1992

[12] H Verma and P Hagedorn ldquoWind induced vibrations oflong electrical overhead transmission line spans a modifiedapproachrdquoWind and Structures vol 8 no 2 pp 89ndash106 2005

[13] B Chen J Zheng and W L Qu ldquoPractical method for wind-resistant design of transmission tower-line system by usingviscoelastic dampersrdquo in Proceedings of the 2nd InternationalConference on Structural Condition Assessment Monitoring andImprovement pp 1028ndash1034 Changsha China 2007

[14] B Chen Y L Xu and W L Qu ldquoEvaluation of atmosphericcorrosion damage to steel space structures in coastal areasrdquoInternational Journal of Solids and Structures vol 42 no 16-17pp 4673ndash4694 2005

[15] MKleiber andTDHienTheStochastic Finite ElementMethodBasic Perturbation Technique and Computer ImplementationJohn Wiley amp Sons New York NY USA 1992

[16] K J BatheFinite Element Procedures Prentice-Hall New JerseyNJ USA 1996

[17] H M Irvine Cable Structure The MIT Press New York NYUSA 1981

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

[18] L Kempner Jr and S Smith ldquoCross-rope transmission tower-line dynamic analysisrdquo Journal of Structural Engineering ASCEvol 110 no 6 pp 1321ndash1335 1984

[19] B Chen W H Guo P Y Li andW P Xie ldquoDynamic responsesand vibration control of the transmission tower-line system astate-of-the-art reviewrdquo The Scientific World Journal vol 2014Article ID 538457 20 pages 2014

[20] A S Pall and C Marsh ldquoResponse of friction damped bracedframesrdquo Journal of the Structural Division vol 108 no 6 pp1313ndash1323 1982

[21] D K Nims P J Richter and R E Bachman ldquoThe use ofthe energy dissipating restraint for seismic hazard mitigationrdquoEarthquake Spectra vol 9 no 3 pp 467ndash489 1993

[22] Y L Xu and B Chen ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part I-methodologyrdquo Engineering Structures vol 30no 7 pp 1789ndash1801 2008

[23] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampers Part II Numerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008

[24] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of