Control of Wind-Induced Vibration of Transmission Tower-Line

Research ArticleControl of Wind-Induced Vibration of TransmissionTower-Line System by Using a Spring Pendulum

Peng Zhang Liang Ren Hongnan Li Ziguang Jia and Tao Jiang

Faculty of Infrastructure Engineering Dalian University of Technology Dalian Liaoning 116024 China

Correspondence should be addressed to Peng Zhang peng1618163com

Received 23 October 2014 Revised 12 February 2015 Accepted 12 February 2015

Academic Editor Massimo Scalia

Copyright copy 2015 Peng Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The high-voltage power transmission tower-line system which is a high flexible structure is very susceptible to the wind-inducedvibrationsThis paper proposes the utilization of the internal resonance feature of the spring pendulum to reduce the wind-inducedvibration of a transmission towerThe kinetic expression of the spring pendulum system is obtained through Lagrangian equationThe condition of the internal resonance is verified to be120582= 2 inwhich120582 is the ratio of the springmode frequency over the pendulummode frequency A 55m tower in the Liaoning province is established in SAP2000 to numerically verify the effectiveness of theproposed device The spring pendulum is modeled using Link element The wind speed history is generated based on Kaimalspectrum using harmonic superposition method Results show that (1) compared with the suspended mass pendulum the springpendulum absorbs more energy and reduces the oscillation more effectively and (2) the vibration control performance of theproposed spring pendulum improves as the external wind load increases

1 Introduction

The high-voltage transmission tower-line system is a vitalfacility in the power transmission system However thetransmission tower-line system generally has low frequen-cies and damping ratios associated with their fundamentaloscillation modes and when subjected to wind loads itmay experience large amplitudes of oscillation [1] Recentinvestigations have reported several tower failures due tostrong wind excitations [2 3] Failures of the transmissiontower-line system may cause massive blackouts resulting inmajor direct and indirect consequences on the economy andnational security [4] It is of great importance to reduce thevibrational response and improve the reliability of powertransmission towers under wind loads

During the past two decades many researchers andengineers have paid much attention to this issue and pro-posed many techniques to improve the wind-resistance ofthe transmission towers [5] Conventional techniques aim atincreasing the stiffness of the transmission tower by enlargingthe cross-section area of the steel members or shorteningthe effective member length by adding additional membersThese methods will tremendously increase the project cost

and thus are not financially acceptable [2] An alternativeapproach to prevent the catastrophic damage of transmissiontower is to install vibration control devices [5 6]

The vibration control devices for vibration control oftransmission towers can be divided into two categoriesOne category aims at increasing the damping ratio byincorporating energy dissipating dampers such as friction-type reinforcingmembers [2 7 8] magnetorheological (MR)dampers [9] viscoelastic dampers (VED) [10] lead viscoelas-tic dampers (LVED) [11 12] and giant magnetostrictivematerial (GMM) actuators [13]

The other category involves attaching a linear dynamicabsorber to the tower Kilroe [14] installed vibrationabsorbers on members of tower arms to mitigate the fatiguephenomena Battista et al [15] proposed a pendulum-likedamper which reached reduction efficiency of over 90with the first mode of vibration Liu and Li [16] combinedthe linear spring model and the Maxwell model to simulatethe tuned mass damper (TMD) in a 3D tower modelNumerical results showed that the vibrational responsewas reduced by 17 with optimal design He et al [17]proposed a suspended mass pendulum (SMP) for vibrationreduction of a transmission tower Both numerical and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 671632 10 pageshttpdxdoiorg1011552015671632

2 Mathematical Problems in Engineering

experimental results verified that the SMP was very effectivein reducing acceleration of the tower These absorbers are alllinear absorbers Introducing nonlinearity may improve thevibration control performance

Recently nonlinear vibration absorbers have been devel-oped in a related field of vibration control of buildings Thenonlinear attachments can localize the vibration energy ina one-way irreversible fashion [18 19] This phenomenonis called nonlinear energy sink (NES) Compared with thetraditional linear dynamic absorber NES has awider effectivefrequency band and better vibration reduction performance[18 19] The NES can be achieved in many ways such asincorporating a nonlinear cubic spring [19] incorporatingcollisions [20] or incorporating the internal resonance [21]

The spring pendulum (SP) which is basically a masssuspended by a spring can be regarded as a combinationof a simple pendulum and a spring oscillator [22 23] Ithas two modes of vibrations the pendulum mode and thespring mode When the SP is correctly designed the twovibration modes will be strongly coupled and vibrationenergy can be transmitted between the two modes Thisis called internal resonance and because of this internalresonance feature the SP can be designed as a new type ofNES for vibration control of power transmission tower-linesystems

In this paper the SP is proposed as a new type of NES toimprove the wind-resistant performance of the transmissiontower-line system In Section 2 the kinetic expression ofthe SP will be obtained through Lagrangian equation Theinternal resonance feature will be discussed and the optimaldesign of a SP will be given In Section 3 a multimassmodel of a practical transmission tower will be establishedto numerically verify the effectiveness of the SP The windfield will be simulated using superposition method based onKaimal spectrum The numerical results will be presented inSection 4 Comparedwith the traditional SMP the SP absorbsmore energy and reduces the oscillation more effectivelyMoreover the intensity of the wind load will be varied toinvestigate its influence on the vibration control performancein this section

2 Mechanism of the Spring Pendulum

21 Mathematical Model of a SP As shown in Figure 1 theSP is basically a mass block suspended by a linear springThe motion equation can be derived through Lagrangianequation In Figure 1 119896

119904denotes the stiffness of the spring

119897ori is the original length of the spring 1198970is the length under

gravity load and119898119901is the mass of the mass blockTherefore

119897ori = 1198970 minus119898119901119892

119896119904

(1)

Establishing a rectangular coordinate at the center of themass block 119909

119901and 119910

119901denote the displacements of the mass

block in the two directions Correspondingly 119901 119901 119901 and

yx

l0lori

yp

xp

mpgks

Figure 1 Schematic of the spring pendulum

119901are the velocity and acceleration The total kinetic energy

and potential energy are

119881 =1

2119896119904(radic119909119901

2 + (1198970+ 119910119901)2

minus 119897ori)

2

minus 119898119901119892119910119901

119879 =1

2119898119901119901

2

+1

2119898119901

2

(2)

Substitute (2) into the Lagrangian equation and themotion equation can be obtained as follows

119901= minus

119896119904

119898119901

119909119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

119901= 119892 minus

119896119904

119898119901

119910119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

(3)

At any moment the forces on the mass block in the twodirections are

119891119909= 119896119904119909119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

119891119910= 119896119904119910119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)minus119898119901119892

(4)

22 Internal Resonance Feature of the SP The SP has twovibration modes the spring mode and pendulum modeDefine 120582 as the ratio of natural frequencies of the twovibrating modes Consider

120582 =120596119904

120596119901

= radic119896119904119898119901

1198921198970

= radic1 +119896119904119897ori

119898119901119892 (5)

Mathematical Problems in Engineering 3

where 120596119904and 120596

119901are the circular frequencies of the spring

mode and pendulum modeAccording to the literature [22] if 120582 = 2 the spring

mode and the pendulum mode will be strongly coupledand the kinetic energy will be transferred between thosetwo modes Figure 2 presents the locus diagram of a SP(119898119901= 00485 kg 119896

119904= 23Nm and 119897ori = 082m) The

locus diagram agrees well with the results of the literature[22] As shown in Figure 2 the suspended mass initiallyswings as a pendulum the kinetic energy is mostly in thependulum mode After several periods the kinetic energyis transmitted to the spring mode and the vibration of themass is mostly in the vertical direction like a simple spring-mass oscillator This phenomenon which is named internalresonance can enlarge the energy absorbing ability of thependulum Therefore SP is more effective in reducing thevibration than a regular SMP

23 One DOF Structure Controlled by a SP Figure 3 shows aone-degree of freedom (DOF) structure controlled by a SP Itsequation of motion can be written as follows

119898 + 119888 + 119896119909 = 119865 + 119891119909 (6)

where119898 119888 and 119896 are the mass damping and stiffness of theprimary structure and 119909 are the acceleration velocityand displacement 119865 denotes the loading force and 119891

119909is the

restoring force generated by the SP 119891119909can be calculated by

(4) Substitute (4) into (6) and (6) can be expressed as follows

119898 + 119888 + 119896119909 + 120572geo119896119904 (119909119901 minus 119909) = 119865 (7)

where

120572geo = (1 minus119897ori

radic(119909119901minus 119909)2

+ (1198970+ 119910119901)2

) (8)

It can be easily verified that 120572geo gt 0 Therefore the SP addeda nonlinear spring to the primary structure This nonlinearspring can reduce the vibration of the primary structure

The vibration control mechanism of the SP can also beexplained in terms of energyThe SPhas two vibrationmodesthe pendulum mode and the spring mode It can absorbthe kinetic energy of the primary structure to the pendulummode thus reducing the vibration of the primary structureIn this case the SP works like a regular SMP Moreover thekinetic energy of the pendulummode can also be transmittedinto the spring mode Therefore the SP has larger energyabsorbing ability than the regular SMP and is more effectivein vibration control than the SMP

24 Optimal Design of the SP Two conditions should besatisfied in order to achieve the optimal design of theSP One condition is that the frequency of the pendulummode matches the frequency of the primary structure to becontrolled that is 119891

119901= 119891primary Here 119891119901 and 119891primary are the

frequency of the spring pendulum and the primary structure

minus02 minus01 0 01 02minus01

minus005

0

005

01

x (m)

y (m

)

Figure 2 Locus of the mass 120582 = 2

k

m

c

xF

yp

xpmp

ks

Figure 3 Schematic of one DOF system controlled by a springpendulum

Figure 4 SZ21-type tangent transmission tower in Liaoningprovince


Span 1 Span 2 Span 3 Span 4

Ground line ConductorsTower 1 Tower 2 Tower 3

400m400m

Figure 5 Transmission tower-line system

Since the frequency of the pendulum mode is determinedby the length of the spring 119897ori 119897ori should be determined asfollows

119897ori =119892

(2120587119891)2 (9)

where 119892 is the gravitational accelerationThe other condition to be satisfied is that 120582 = 2 Then (5)

can be written as follows

120582 = radic1 +119896119904119897ori

119898119901119892= 2 (10)

Therefore

119896119904=3119898119901119892

119897ori (11)

where 119896119904is the stiffness of the spring 119898

119901is the mass of the

SP and 119897ori is the length of the spring

3 Numerical Model of the TransmissionTower-Line System Controlled by SP

31 Prototype and the FEM Model of the Transmission Tower-Line System In order to verify the vibration control effective-ness of the proposed SP a practical transmission tower-linesystem illustrated in Figures 4 and 5 is chosen as the primarystructure to be controlledThe tangent tower is constructed ofQ345 angle steels Its height is 539mThepower transmissiontower-line system includes three towers and four span linesas shown in Figure 4 The distance between adjacent towersis 400m The upper 8 cables are ground lines and lower 24cables are four-bundled conductor lines The properties ofthe conductor line and ground line properties are listed inTable 1

The transmission tower-line system is established ina FEM software SAP2000 The angle steels of the towerand the isolators are modeled using Beam element theconductorground lines are modeled using Cable elementThe base points of the transmission tower are fixed on theground The connections between transmission towers andlines are hinged

Angle steelof the tower

Hinged to the tower

The link element

Assign the massto the joint

Figure 6 Simulating the SP using Link element in SAP2000

Table 1 Conductor line and ground line properties

Conductor line Ground lineType LGJ-40035 LGJ-9555Outside diameter (mm) 2682 16Modulus (GPa) 65 105Cross-section (mm2) 42524 15281Mass per unit length (Kgm) 13490 06967

32 Modeling the SP in SAP2000 The SP is simulated byLink element in SAP2000 as shown in Figure 6 Length andstiffness of the Link element are calculated using (9) and (11)One joint of the Link element is hinged to the tower to becontrolled Mass of the SP is assigned to the other joint of theLink element Figure 7 shows the locus of a spring pendulumcalculated by SAP2000 The curve attained by SAP2000 is ingood accordance with the theoretical results Therefore theLink element can be used to simulate the behavior of the SP

33 Simulation of Wind Field In wind engineering the totalwind velocity can be simulated as the sum of the meanvelocity and the fluctuating wind velocity

V (119911 119905) = V (119911) + V119891(119911 119905) (12)


minus02 minus01 0 01 0201

005

0

minus005

minus01

Theoretical resultsSAP2000 results

x (m)

y (m

)

Figure 7 Locus of a SP calculated by SAP2000

where V(119911 119905) is the total wind velocity V(119911) is the mean windvelocity and V

119891(119911 119905) denotes the fluctuating wind velocity

According to the statistic data the mean wind is oftenexpressed by logarithmic function or exponent functionassociated with the height [24]

V (119911) =1

119896119906n ln(

119911

1199110

) (13)

where V(119911) is the mean wind speed of height 119911 119896 is Karmanconstant 119911

0is the ground roughness length and 119906n is friction

velocityThe time history of the fluctuating wind can be regarded

as a random process with a specific power spectrum Accord-ing to the theory presented by Shinozuka and Jan [25] thefluctuating wind can be simulated using harmony superposi-tion method

119906119894(119905) = radic2Δ120596

119894

sum

119897=1

119873

sum

119896=1

10038161003816100381610038161003816119867119894119895(120596119896)10038161003816100381610038161003816cos [120596

119896119905 minus 120579119894119897(120596119896) + 120593119897119896]

119894 = 1 2 119898

(14)

where 119906119894(119905) is the time history of the fluctuating wind Δ120596

is the frequency increment calculated by Δ120596 = 120596119906119901119873 in

which 119873 denotes the division number and 120596119906119901

is the cut-off frequency of calculation 119867

119894119895is obtained from Cholescky

decomposition of the wind cross-spectral density matrix 120579119894119897

is the phase angle of structure between the two different loadpoints 120593

119897119896denotes the random phase distributed in the range

of [0 2120587]Based on the central limitation theorem the random

process 119906119894(119905) will gradually tend to be the Gauss process

when 119873 rarr +infin And the time interval should meet therequirement of Δ119905 le 21205872120596

119906119901[26] In this paper they are

9000

4500

4500

3000

3000

2400

2800

2700

2700

2800

2800

2600

2600

2500

6000

Figure 8 Wind simulation points

selected as 119873 = 8192 Δ119905 = 005 and 120596119906119901

= 60 The targetpower spectrum is Kaimal fluctuating wind power spectrum

In order to study the influence of the wind load intensityon the vibration control performance of the SP the meanwind velocity at the height of 10 meters above the ground119881

10

is selected as 20ms 30ms and 40msBecause the transmission tower nodes are numerous it

is difficult to simulate the wind velocity history for everynode of the tower Consequently the transmission toweris simplified by 15 regions as shown in Figure 8 The windload acts only on the center of each region The position ofeach region center and windward area are listed in Table 2Similarly each cable is divided into 10 segments Wind loadonly acts on the middle of each segment Figure 9 shows


0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)

Figure 9 Time history of wind speed at the top of Tower 2 11988110=

40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

Figure 10 Power spectral comparison between simulated andtheoretical wind samples

the wind speed on the top of Tower 2 at 11988110= 40ms case

Figure 10 compares the target wind power spectrum and thesimulated time history of the fluctuating wind As shown inFigure 10 the simulation results agree well with the targetspectrum

34 Damping Consideration Determination of the dampingratio of the power transmission tower-line system is asophisticated problem Damping ratio of the steel structure isusually valued 2 [27] Moreover the aerodynamic dampingof the conductor has a significant effect on the dynamicperformance [28] According to the theory presented byGani and Legeron [29] the aerodynamic damping can becalculated with the following equation

119888 (119911) = 120588119886119862119889V (119911) 119860 (15)

where 119888(119911) is the aerodynamic damping 120588119886is the intensity of

air 119862119889is the drag coefficients V(119911) is the mean wind speed at

height 119911 and 119860 is the area exposed to wind loading

4 Effectiveness of Vibration Reduction

41 Criteria to Evaluate Vibration Control Performance Inorder to verify the effectiveness of the proposed device a SPand a SMP of the same mass ratio are designed to compare

450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)

Without controlWith SMPWith SP

Figure 11 Displacement time history at the top of Tower 2 11988110=

40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5


Figure 12 Acceleration time history at the top of Tower 2 11988110=

40ms

Table 2 The position of wind load and windward area

Region Height of theregion center (m)

Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611

their vibration control effectiveness Here the mass ratio isdefined as follows

120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60

Axial force of angle steel (kN)

Hei

ght (

m)


(c)

Figure 13 Envelope of (a) displacement (b) acceleration and (c) axial force of the 11988110= 40ms case


Table 3 Comparison of the vibration reduction ratio of SP and SMP

Wind speed Max disp Max acc Max axial force RMS of disp RMS of accWith SP SMP SP SMP SP SMP SP SMP SP SMP

11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184

where 120583 is the mass ratio 119898 is the mass of the vibrationcontrol device and119872 is the mass of the power transmissiontower In this paper 120583 is selected to be 2

To evaluate the vibration control effectiveness of the SPand the SMP the vibration reduction ratio is defined asfollows

120578 =1199030minus 119903ctrl1199030

times 100 (17)

where 120578 is the vibration reduction ratio and 1199030and 119903ctrl

are the response of the tower with and without vibrationcontrol devices 119903

0and 119903ctrl can be the maximum value or the

root mean square value (RMS) The maximum value can becalculated with the following equation [28 30]

119903 = 119903 + 119892119903 (18)

where 119903 is the peak dynamic response 119903 is the mean or timeaverage response 119903 is the RMS of the fluctuating responseand 119892 is the gust response factor generally in the range of 3-4[28]

42 Comparison of Vibration Reduction Performance Basedon the numerical model and the simulated wind fielddynamic response of the power transmission tower-linesystem can be calculated The time step is chosen as 005 sFigures 11ndash13 show the results of the 119881

10= 40ms case

Results of 11988110= 20ms case and 119881

30= 30ms case are very

similar to that of the11988110= 40ms case Therefore the figures

of those two cases are omitted in this paper The vibrationreduction ratios of those two cases can be found in Table 3

Figures 11 and 12 compare the displacement and acceler-ation at the top of Tower 2 from 450 s to 500 s in the 119881

10=

40ms case Figure 13 shows the envelopes of displacementsaccelerations and internal forces of Tower 2 It is clearthat SP is more effective than the SMP The SP effectivelyreduced the RMS value of the displacement and accelerationby 361 and 302 whereas that of the SMP is only 286and 184 respectively The proposed SP mitigated themaximum displacement of Tower 2 by 151 larger than thatof the SMP which is only 121 In terms of accelerationgreater effective mitigation is achieved by the SP having themaximum value reduced by 193The reduction ratio of theSMP is only 138 In terms of axial internal force greatereffective performance has also been observed with the SP InFigure 13(c) the largest axial internal force is reduced by theSP by 202 which is also larger than the reduction ratio ofSMP which is 110

43 Influence of the Wind Speed on the Vibration ReductionPerformance The SP is essentially a nonlinear damper Thusits vibration control performance may be influenced bythe intensity of external loads Table 3 lists the vibrationreduction ratio of the SP under 3 cases namely the 119881

10=

20ms case the 11988110

= 30ms case and the 11988110

= 40mscase It is clear from Table 3 that the vibration performanceis influenced by the level of wind loads As the windspeed increases the vibration reduction ratio of the SP alsoincreases which implies that the SP is more effective understrong winds

This phenomenon may be explained this way at lowwind speed the vibration of the transmission tower is smallTherefore the SP absorbs little kinetic energy from thetransmission tower and itsmotion is very smallThe vibrationof the SP ismostly in the pendulummode Figure 14(a) showsthe locus of the SP in the119881

10= 20ms As can be seen the SP

vibrates like a linear SMP Therefore its vibration reductionratio is close to that of the SMP As the wind speed increasesthe SP will vibrate more severely as it absorbs more energyFigure 14(b) corresponds to the case of 119881

10= 40ms As

can be seen the motion of the SP is larger than that of the11988110= 20ms case and the locus is no longer similar to that

of the SMP This implies that more energy is transmitted tothe spring mode of the SP The vibration absorbing ability isimproved by the internal resonance feature

5 Conclusion

This paper proposed to utilize the internal resonance featureof the SP in order to reduce the wind-induced vibration ofthe power transmission tower-line systemThemotion equa-tion of the spring pendulum system was obtained throughLagrangian equation A practical transmission tower-linesystem was established in SAP2000 to assist numerical studyAn SP and an SMP with the same mass ratio were designedto compare the vibration control performance Based ontheoretical analysis and numerical results the followingconclusions are drawn

(1) The condition of internal resonance is verified to be120582 = 2 When this condition is satisfied the pendulummode and the spring mode of the SP will be stronglycoupled and the vibration energy will be transmittedfrom the pendulum mode to spring mode Conse-quently the SP has larger vibration absorbing abilitythan the regular SMPand ismore effective in reducingthe vibration of the primary structure


minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)

minus003 minus002 minus001 0 001 002

minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)

Figure 14 Locus of the SP (a) the 11988110= 20ms case and (b) the 119881

10= 40ms case

(2) The proposed SP can be simulated using the Linkelement in SAP2000The numerical results calculatedby SAP2000 agree well with the theoretical results

(3) The numerical study verified that the SP showedbetter vibration control performance than the regularSMP

(4) The vibration control performance of the SP is influ-enced by the wind load The SP works like a regularSMP and its vibration reduction ratio is small atlow wind speed As the wind speed increases theinternal resonance feature of the SP becomes moreobvious and the vibration reduction ratio increasesThis implies that the SP is more effective under largerwind loads

Conflict of Interests

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

Acknowledgment

This research work was supported by the Science Fund forCreative Research Groups of the National Natural ScienceFoundation of China (Grant no 51121005)

References

[1] Z Zhang H Li G Li W Wang and L Tian ldquoThe numer-ical analysis of transmission tower-line system wind-inducedcollapsed performancerdquoMathematical Problems in Engineeringvol 2013 Article ID 413275 11 pages 2013

[2] J-H Park B-W Moon K-W Min S-K Lee and C KKim ldquoCyclic loading test of friction-type reinforcing members

upgrading wind-resistant performance of transmission towersrdquoEngineering Structures vol 29 no 11 pp 3185ndash3196 2007

[3] Q Xie Y Zhang and J Li ldquoInvestigation on tower collapses of500 kV renshang 5237 transmission line caused by downburstrdquoPower System Technology vol 2006 pp 59ndash63 89 2006

[4] 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

[5] B ChenW-HGuo 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

[6] C-X Li J-H Li and Z-Q Yu ldquoA review of wind-resistantdesign theories of transmission tower-line systemsrdquo Journal ofVibration and Shock vol 28 no 10 pp 15ndash25 2009

[7] J P Wang B Chen and S M Sun ldquoSeismic response controlof transmission tower-line system with friction dampersrdquo inProceedings of the 6th International Conference on Advances inSteel Structures and Progress in Structural Stability and Dynam-ics (ICASS rsquo09) pp 1137ndash1142 Hong Kong China December2009

[8] B ChenW-L Qu and J Zheng ldquoSemi-active control for wind-induced responses of transmission tower-line system usingfriction dampersrdquo EngineeringMechanics vol 26 no 1 pp 221ndash226 2009

[9] 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

[10] W Wang D Zhao W Zhong J Xu J Ling and X XiaoldquoEffect of wind-induced vibrationrsquos control for high-voltagetransmission tower based on prototype measurementrdquo Journalof Central South University (Science and Technology) vol 44 pp3911ndash3917 2013


[11] L Li H Cao K Ye and Y Jiang ldquoSimulation of galloping andwind-induced vibration controlrdquo Noise and Vibration World-wide vol 41 no 10 pp 15ndash21 2010

[12] L Li and P Yin ldquoThe research on wind-induced vibrationcontrol for big-span electrical transmission tower-line systemrdquoGongcheng LixueEngineering Mechanics vol 25 no supple-ment 2 pp 213ndash229 2008

[13] S LWang X Y Zhu J Q Zhu J B Dai and X Zhao ldquoOptimalplacement of GMM actuators for active vibration control of apower transmission towerrdquo Journal of Vibration and Shock vol31 no 19 pp 48ndash52 2012

[14] N Kilroe ldquoAerial method to mitigate vibration on transmissiontowersrdquo in Proceedings of the IEEE 9th International Confernceon Trasmission and Distribution Construction Operation andLive-Line Maintenance Proceedings (ESMO rsquo00) pp 187ndash194Montreal Canada 2000

[15] 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

[16] G H Liu and H N Li ldquoAnalysis and optimization control ofwind-induced dynamic response for high-voltage transmissiontower-line systemrdquo Proceedings of the Chinese Society of Electri-cal Engineering vol 28 no 19 pp 131ndash137 2008

[17] Y He W Lou B Sun S Yu Y Guo and Y Ye ldquoWind-induced vibration control of long span transmission tower withsuspended mass pendulumsrdquo Journal of Zhejiang University(Engineering Science) vol 39 pp 1891ndash1896 2005

[18] A F Vakakis ldquoInducing passive nonlinear energy sinks invibrating systemsrdquo Journal of Vibration and Acoustics Transac-tions of the ASME vol 123 no 3 pp 324ndash332 2001

[19] X Jiang D Michael Mcfarland L A Bergman and A FVakakis ldquoSteady state passive nonlinear energy pumping incoupled oscillators theoretical and experimental resultsrdquo Non-linear Dynamics vol 33 no 1 pp 87ndash102 2003

[20] P Zhang G Song H-N Li and Y-X Lin ldquoSeismic controlof power transmission tower using pounding TMDrdquo Journal ofEngineering Mechanics vol 139 no 10 pp 1395ndash1406 2013

[21] Y-C Zhang X-R Kong Z-X Yang and H-L Zhang ldquoTar-geted energy transfer and parameter design of a nonlinearvibration absorberrdquo Journal of Vibration Engineering vol 24 no2 pp 111ndash117 2011

[22] Z B Yang QH Xia and S P Liu ldquoThe spring pendulumunderdifferent controlled parametersrdquo College Physics no 5 pp 23ndash26 42 2011

[23] J P Wang H Z Liu D N Yuan and Z X Xu ldquoAn improvedstate space model for parts of nonlinear oscillation systems andits numerical methodrdquo Chinese Journal of Applied Mechanicsvol 112 pp 116ndash167 2002

[24] X Zhang Handbook of Wind Load Thories and Wind-ResistantComputations for Engineering Structures Tongji UniversityPress Shanghai China 1990

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

[26] H-N Li T-H Yi Q-Y Jing L-S Huo and G-X WangldquoWind-induced vibration control of Dalian international trademansion by tuned liquid dampersrdquo Mathematical Problems inEngineering vol 2012 Article ID 848031 21 pages 2012

[27] J Yang F Yang Q Li D Fu and Z Zhang ldquoDynamic responsesanalysis and disaster prevention of transmission line under

strong windrdquo in Proceedings of the Interntional Conference onPower System Technology pp 1ndash6 IEEE October 2010

[28] A M Loredo-Souza and A G Davenport ldquoThe effects of highwinds on transmission linesrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 74-76 pp 987ndash994 1998

[29] FGani andF Legeron ldquoDynamic response of transmission linesguyed towers under wind loadingrdquo Canadian Journal of CivilEngineering vol 37 no 3 pp 450ndash464 2010

[30] H Yasui H Marukawa Y Momomura and T Ohkuma ldquoAna-lytical study on wind-induced vibration of power transmissiontowersrdquo Journal of Wind Engineering amp Industrial Aerodynam-ics vol 83 pp 431ndash441 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


experimental results verified that the SMP was very effectivein reducing acceleration of the tower These absorbers are alllinear absorbers Introducing nonlinearity may improve thevibration control performance

Recently nonlinear vibration absorbers have been devel-oped in a related field of vibration control of buildings Thenonlinear attachments can localize the vibration energy ina one-way irreversible fashion [18 19] This phenomenonis called nonlinear energy sink (NES) Compared with thetraditional linear dynamic absorber NES has awider effectivefrequency band and better vibration reduction performance[18 19] The NES can be achieved in many ways such asincorporating a nonlinear cubic spring [19] incorporatingcollisions [20] or incorporating the internal resonance [21]

The spring pendulum (SP) which is basically a masssuspended by a spring can be regarded as a combinationof a simple pendulum and a spring oscillator [22 23] Ithas two modes of vibrations the pendulum mode and thespring mode When the SP is correctly designed the twovibration modes will be strongly coupled and vibrationenergy can be transmitted between the two modes Thisis called internal resonance and because of this internalresonance feature the SP can be designed as a new type ofNES for vibration control of power transmission tower-linesystems

In this paper the SP is proposed as a new type of NES toimprove the wind-resistant performance of the transmissiontower-line system In Section 2 the kinetic expression ofthe SP will be obtained through Lagrangian equation Theinternal resonance feature will be discussed and the optimaldesign of a SP will be given In Section 3 a multimassmodel of a practical transmission tower will be establishedto numerically verify the effectiveness of the SP The windfield will be simulated using superposition method based onKaimal spectrum The numerical results will be presented inSection 4 Comparedwith the traditional SMP the SP absorbsmore energy and reduces the oscillation more effectivelyMoreover the intensity of the wind load will be varied toinvestigate its influence on the vibration control performancein this section

2 Mechanism of the Spring Pendulum

21 Mathematical Model of a SP As shown in Figure 1 theSP is basically a mass block suspended by a linear springThe motion equation can be derived through Lagrangianequation In Figure 1 119896

119904denotes the stiffness of the spring

119897ori is the original length of the spring 1198970is the length under

gravity load and119898119901is the mass of the mass blockTherefore

119897ori = 1198970 minus119898119901119892

119896119904

(1)

Establishing a rectangular coordinate at the center of themass block 119909

119901and 119910

119901denote the displacements of the mass

block in the two directions Correspondingly 119901 119901 119901 and

yx

l0lori

yp

xp

mpgks

Figure 1 Schematic of the spring pendulum

119901are the velocity and acceleration The total kinetic energy

and potential energy are

119881 =1

2119896119904(radic119909119901

2 + (1198970+ 119910119901)2

minus 119897ori)

2

minus 119898119901119892119910119901

119879 =1

2119898119901119901

2

+1

2119898119901

2

(2)

Substitute (2) into the Lagrangian equation and themotion equation can be obtained as follows

119901= minus

119896119904

119898119901

119909119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

119901= 119892 minus

119896119904

119898119901

119910119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

(3)

At any moment the forces on the mass block in the twodirections are

119891119909= 119896119904119909119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)

119891119910= 119896119904119910119901(1 minus

119897ori

radic119909119901

2 + (1198970+ 119910119901)2

)minus119898119901119892

(4)

22 Internal Resonance Feature of the SP The SP has twovibration modes the spring mode and pendulum modeDefine 120582 as the ratio of natural frequencies of the twovibrating modes Consider

120582 =120596119904

120596119901

= radic119896119904119898119901

1198921198970

= radic1 +119896119904119897ori

119898119901119892 (5)


where 120596119904and 120596




119904= 23Nm and 119897ori = 082m) The



119898 + 119888 + 119896119909 = 119865 + 119891119909 (6)


119909is the



119898 + 119888 + 119896119909 + 120572geo119896119904 (119909119901 minus 119909) = 119865 (7)

where

120572geo = (1 minus119897ori

radic(119909119901minus 119909)2

+ (1198970+ 119910119901)2

) (8)







minus005

0

005

01

x (m)

y (m

)


k

m

c

xF

yp

xpmp

ks






400m400m



119897ori =119892

(2120587119891)2 (9)



120582 = radic1 +119896119904119897ori

119898119901119892= 2 (10)

Therefore

119896119904=3119898119901119892

119897ori (11)








Hinged to the tower

The link element







V (119911 119905) = V (119911) + V119891(119911 119905) (12)



005

0

minus005

minus01


x (m)

y (m

)





V (119911) =1

119896119906n ln(

119911

1199110

) (13)





119906119894(119905) = radic2Δ120596

119894

sum

119897=1

119873

sum

119896=1

10038161003816100381610038161003816119867119894119895(120596119896)10038161003816100381610038161003816cos [120596

119896119905 minus 120579119894119897(120596119896) + 120593119897119896]

119894 = 1 2 119898

(14)













9000

4500

4500

3000

3000

2400

2800

2700

2700

2800

2800

2600

2600

2500

6000


selected as 119873 = 8192 Δ119905 = 005 and 120596119906119901



10




0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)


40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3





119888 (119911) = 120588119886119862119889V (119911) 119860 (15)






450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)



40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5



40ms



Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611


120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120596119904and 120596




119904= 23Nm and 119897ori = 082m) The



119898 + 119888 + 119896119909 = 119865 + 119891119909 (6)


119909is the



119898 + 119888 + 119896119909 + 120572geo119896119904 (119909119901 minus 119909) = 119865 (7)

where

120572geo = (1 minus119897ori

radic(119909119901minus 119909)2

+ (1198970+ 119910119901)2

) (8)







minus005

0

005

01

x (m)

y (m

)


k

m

c

xF

yp

xpmp

ks






400m400m



119897ori =119892

(2120587119891)2 (9)



120582 = radic1 +119896119904119897ori

119898119901119892= 2 (10)

Therefore

119896119904=3119898119901119892

119897ori (11)








Hinged to the tower

The link element







V (119911 119905) = V (119911) + V119891(119911 119905) (12)



005

0

minus005

minus01


x (m)

y (m

)





V (119911) =1

119896119906n ln(

119911

1199110

) (13)





119906119894(119905) = radic2Δ120596

119894

sum

119897=1

119873

sum

119896=1

10038161003816100381610038161003816119867119894119895(120596119896)10038161003816100381610038161003816cos [120596

119896119905 minus 120579119894119897(120596119896) + 120593119897119896]

119894 = 1 2 119898

(14)













9000

4500

4500

3000

3000

2400

2800

2700

2700

2800

2800

2600

2600

2500

6000


selected as 119873 = 8192 Δ119905 = 005 and 120596119906119901



10




0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)


40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3





119888 (119911) = 120588119886119862119889V (119911) 119860 (15)






450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)



40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5



40ms



Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611


120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












400m400m



119897ori =119892

(2120587119891)2 (9)



120582 = radic1 +119896119904119897ori

119898119901119892= 2 (10)

Therefore

119896119904=3119898119901119892

119897ori (11)








Hinged to the tower

The link element







V (119911 119905) = V (119911) + V119891(119911 119905) (12)



005

0

minus005

minus01


x (m)

y (m

)





V (119911) =1

119896119906n ln(

119911

1199110

) (13)





119906119894(119905) = radic2Δ120596

119894

sum

119897=1

119873

sum

119896=1

10038161003816100381610038161003816119867119894119895(120596119896)10038161003816100381610038161003816cos [120596

119896119905 minus 120579119894119897(120596119896) + 120593119897119896]

119894 = 1 2 119898

(14)













9000

4500

4500

3000

3000

2400

2800

2700

2700

2800

2800

2600

2600

2500

6000


selected as 119873 = 8192 Δ119905 = 005 and 120596119906119901



10




0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)


40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3





119888 (119911) = 120588119886119862119889V (119911) 119860 (15)






450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)



40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5



40ms



Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611


120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











005

0

minus005

minus01


x (m)

y (m

)





V (119911) =1

119896119906n ln(

119911

1199110

) (13)





119906119894(119905) = radic2Δ120596

119894

sum

119897=1

119873

sum

119896=1

10038161003816100381610038161003816119867119894119895(120596119896)10038161003816100381610038161003816cos [120596

119896119905 minus 120579119894119897(120596119896) + 120593119897119896]

119894 = 1 2 119898

(14)













9000

4500

4500

3000

3000

2400

2800

2700

2700

2800

2800

2600

2600

2500

6000


selected as 119873 = 8192 Δ119905 = 005 and 120596119906119901



10




0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)


40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3





119888 (119911) = 120588119886119862119889V (119911) 119860 (15)






450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)



40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5



40ms



Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611


120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 50020

40

60

80

Time (s)

Win

d sp

eed

(ms

)


40ms

Frequency (Hz)

Pow

er sp

ectr

um

Target spectrum

Simulation results

102

101

100

100

10minus1

10minus1

10minus2

10minus2

10minus3





119888 (119911) = 120588119886119862119889V (119911) 119860 (15)






450 455 460 465 470 475 480 485 490 495 5000

005

01

015

Time (s)

Disp

lace

men

t (m

)



40ms

450 455 460 465 470 475 480 485 490 495 500

0

5

Time (s)Ac

cele

ratio

n (m

s2)

minus5



40ms



Windwardarea (m2)

1 900 70832 150 45073 195 18684 240 24355 268 23696 300 23447 324 20298 352 16729 379 160210 406 181011 434 159112 462 111713 488 106214 514 116415 539 0611


120583 =119898

119872times 100 (16)


0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 001 002 0030

10

20

30

40

50

60

Displacement (m)

Hei

ght (

m)


(a)

0 2 4 60

10

20

30

40

50

60

Hei

ght (

ms

)


Acceleration (ms2)

(b)

0 50 100 1500

10

20

30

40

50

60


Hei

ght (

m)


(c)





11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11988110= 20ms 127 100 94 102 126 121 309 249 228 142

11988110= 30ms 133 129 147 116 149 106 350 278 278 160

11988110= 40ms 151 121 193 138 202 110 361 286 302 184



120578 =1199030minus 119903ctrl1199030

times 100 (17)





119903 = 119903 + 119892119903 (18)



10= 40ms case






10=



10=







10= 40ms As



5 Conclusion




minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus001 minus0005 0 0005 001

minus2

minus1

0

1

2

3

(m)

(m)

SMPSP

times10minus3

(a)


minus5

0

5

10

(m)

(m)

SMPSP

times10minus3

(b)


10= 40ms case






Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Control of Wind-Induced Vibration of Transmission Tower-Line