Review ArticleClosed-Form Formula of the Transverse Dynamic Stiffness ofa Shallowly Inclined Taut Cable

Dan-hui Dan1 Zu-he Chen1 and Xing-fei Yan2

1 Department of Bridge Engineering Room 711 Bridge Building Tongji University 1239 Siping Road Shanghai 200092 China2 Shanghai Urban Construction Design Research Institute Shanghai 200125 China

Correspondence should be addressed to Dan-hui Dan dandanhuitongjieducn

Received 1 December 2013 Revised 30 April 2014 Accepted 6 May 2014 Published 28 May 2014

Academic Editor Mickael Lallart

Copyright copy 2014 Dan-hui Dan 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 segmented vibration-governed equations and their general solutions for cables acted upon by intermediate transverse forcesare derived by applying Hamiltonrsquos principle Including the effects of sagging flexible stiffness clamped boundary conditionsand inclination angle of the cable the element-wise dynamic stiffness for each cable segment split into segments having uniquetransverse forces is derived By using methods from the global stiffness assembly process of FEM the global level of the cablesrsquodynamic equilibrium equation is obtained and as a result the final closed-form formula of transverse dynamic stiffness is derivedAdditionally the corresponding analytic form without considering sagging effects is also obtained Case studies are conducted onthe aspects of accuracy rationality of the distribution on the spatial field and frequency domains of dynamic stiffness calculationsBy comparison with the Guyan-based static FEM reduction method it is shown that the result obtained from the proposed closed-form solution which includes sagging effects is exact and rational thus creating a powerful tool in transverse vibration analysis

1 Introduction

A current trend in structure development is exceeding exist-ing spanning limits for which cable structures are playing anincreasingly important role With the extensive use of cablestructures the use of lateral bracing is becoming commonincluding the installation of intermediate turn supportslateral dampers and the use of cross ties chain bar betweencables and hanger rods [1 2] These elements provide trans-verse static forces for the main body of the cable structuresand are often specially designed for improving the dynamiccharacteristics of the cable structures [3 4] The analysis andcalculation of characteristics and behaviors of cables actedupon by transverse forces exerted by these elements are alsoimportant factors in the design of a cable system A directrelationship between the excitation and vibration responsecan be determined by an analysis of the dynamic stiffnessand therefore this particular analysis method has become animportant tool in cable vibration analysis [5 6]

Due to the nonlinear behavior of cables the study on theirdynamic stiffness is limited Currently most studies on the

dynamic stiffness of cables are focused on the developmentof a computation method using a 4 times 4 matrix that describesthe two-dimensional dynamic stiffness of one end of a cablewhile the other end is kept fixed Kolousek first providedthe series solution to the dynamic stiffness at the free end ofsuch cables neglecting internal damping [7] Davenport [8](1959) found a closed-formdynamic stiffnessmatrix solutiontaking damping into account and later in 1965 developed theseries solution of dynamic stiffness which also considereddamping effects [9] In 1981 on the basis of the closed-formsolution given by Davenport Irvine proposed a simplifiedexpression for dynamic stiffness [1] All of these studiesassumed small vibration amplitudes and a small inclinationangle for the cables In 1983 Veletsos and Darbrel studied amore accurate closed-form solution to the dynamic stiffnessin the upper part of a cable under harmonic excitation atall angles and under a parabolic initial configuration Whendamping effects were not taken into account this closed-formsolution was consistent with the series solution proposedby Veletsos and Darbre [10 11] In 1991 Starossek derived adynamic stiffness matrix for cable structures with inclined

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 497670 14 pageshttpdxdoiorg1011552014497670

2 Shock and Vibration

hinges which took sagging into consideration and analyzedthe features of its elements [12 13] Sarkar and Manoharproposed a calculation scheme for the dynamic stiffnessmatrix of cables using finite element analysis [14] In 2001Kim and Chang gave an approximate analytical expressionof the dynamic stiffness matrix of an inclined cable with acatenary configuration [15] which surpassed the limitationsof the parabolic static configuration

Each of these studies on dynamic stiffness consideredvarious factors based on the taut string theory while theterms corresponding to the bending stiffness of the cableswere all neglected in the passive vibration control equationsThe actual cable structures are usually inclined cables witheach end clamped and with relatively high bending stiffness[16 17] With increased length the effect of sag cannot beneglected [18 19] A calculation algorithm of the dynamicstiffness of the cables with all of these factors taken intoaccount would be very valuable to engineering practicesFurthermore dynamic stiffness is used to describe thedynamical behavior at the ends of cables and thereforecannot be used in the case of a cable acted upon by transverseforces However in engineering practices a majority of cablesystems are acted upon by transverse forces and thereforethe study of transverse dynamic stiffness of cables is crucialand will contribute substantially to studies involving cabledamping systems [20 21]

In this study a cable is modeled as being divided intovarious segments each acted upon by a unique transverseforce while factors including sagging bending stiffnessclamped boundary conditions and inclination angle are allconsideredThe dynamic stiffness method is used to establishan equation of motion and its general solution correspondsto a cable system acted upon by transverse forces Thedynamic displacement of each segment is used to characterizethe dimensionless vibration mode function Eventually ananalytical expression for the transverse dynamic stiffnessmatrix of a cable is given in a clear simple form The resultsof the study lay a foundation for further research in systemvibration

2 General Solution forGoverning Equation of Cable

The focus of this study is on a commonly observed situationin engineering namely when one end of a cable is acted uponby transverse forces As shown in Figure 1 due to the presenceof transverse forces the cable system can be discretized intocable segment 1 (from end a to end c) cable segment 2(from end c to end b) and transverse force elements (suchas dampers)

As shown in Figure 1 by assuming that transverse forcesdonot affect the static configuration of the cable the two cablesegments can be described by the same quadratic parabolicfunction 119910(119909

0) when the cable vibrates segments 1 and 2

follow different configurations namely 119906(1199091 119905) and 119906(119909

2 119905)

When considering the effects of sagging bending stiffnessclamped boundary conditions and inclination angle andignoring the internal damping and shear forces following

H

l0

l1

l2

x0

x1

x2

d

y(x0) = y(x2 + l1)

120579

y(x0) 120597u(x2 t)

120597x2dx2

dx2

a

b

c

d(Δl) asymp ( 120597u(x2 t)120597x2minusdy(x0)

dx0) middot dy(x0)

dx0dx2

u(x2 t)

u(x2 t)

u(x1 t)

Figure 1 Schematic diagram of calculation of additional dynamicstress

governing equation [18 22] can be used to describe the freevibration of the two segments

119864119868

1205974119906

1205971199094

119895

minus 119867

1205972119906

1205971199092

119895

minus ℎ119895

1198892119910

11988911990902+ 119898

1205972119906

1205971199052= 0 (1)

where 119895 = 1 2 119864119868 is the flexural rigidity 119867 is the initialtension force 119898 is the mass per unit length 119910 is theconfiguration of the cable under static equilibrium (assumingthat dampers have no impact on the static configuration)119909119895is the coordinate of each segment and ℎ

119895represents the

additional segmented force which is an additional cabletension generated due to the difference from the staticconfiguration during vibration and is equal to the product ofthe additional strain 120576

119895

V(119905) generated during difference from

the static equilibrium position and the axial stiffnessThat is

ℎ119895= 119864119860120576

119895

V(119905) (2)

The additional dynamic strain 120576119895

V(119905) can be calculated

using the ratio between the change in cable lengthΔ119897119895

V duringvibration and the static cable length 119897

119895

119890 Consider

120576119895

V(119905) =

Δ119897119895

V

119897119895

119890 (3)

In (2) and (3) the superscript V represents the dynamicconfiguration and 119890 represents static equilibrium A similarrule is followed below

As shown in Figure 1 in bridge projects the anglebetween the tangential direction and the chord of a cable isvery small for both the static and dynamic configurationsTherefore when ignoring the higher order terms of the staticand dynamic configurations the amount of elongation 119889119909

119895of

Shock and Vibration 3

a differential segment during vibration relative to that of thestatic configuration 119889(Δ119897

119895

V) can be approximated by

119889 (Δ119897119895

V) asymp (

120597119906 (119909119895 119905)

120597119909119895

minus

119889119910 (1199090)

1198891199090

) sdot

119889119910 (1199090)

1198891199090

119889119909119895

asymp

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895

(4)

The total elongation of segment 119895 can be written as

Δ119897119895

V= int

119897119895

0

119889 (Δ119897119895

V) = int

119897119895

0

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895 (5)

With the assumption of a parabolic static equilibriumone can use the following expression for the configuration

119910 (1199090) = minus

4119890

1198970

1199090(1199090minus 1198970)

= minus

4119890

1198970

(1199092+ 1198971) (1199092minus 1198972)

= minus

4119890

1198970

1199091(1199091minus 1198970)

(6)

where 119890 = 1198891198970= 119898119892119897

0cos 1205798119867 is the sag-span ratio and

119889 = 1198981198921198970

2 cos 1205798119867Starting from (6) and noting the relationship of 119909

0= 1199091=

1199092+1198971 we can easily get the coordinate derivative of the static

configuration function 119910(1199090) as follows

119889119910 (1199090)

1198891199090

= minus

4119890

1198970

[21199090minus 1198970] = minus

4119890

1198970

[21199091minus 1198970]

= minus

4119890

1198970

[21199092+ (1198971minus 1198972)]

(7)

And then by substituting it into (5) we have

Δ1198971

V= int

1198971

0

119889 (Δ1198971

V) = int

1198971

0

120597119906 (1199091 119905)

1205971199091

119889119910 (1199090)

1198891199090

1198891199091

= minus

4119890

1198970

int

1198971

0

120597119906 (1199091 119905)

1205971199091

[21199091minus 1198970] 1198891199091

= minus

4119890

1198970

2int

1198971

0

120597119906 (1199091 119905)

1205971199091

11990911198891199091minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2int

1198971

0

120597

1205971199091

[119906 (1199091 119905) 1199091] 1198891199091

minus2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2[119906 (1199091 119905) 1199091]1198971

0minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1

minus 1198970119906(1199091 119905)1198971

0

= minus

4119890

1198970

21198971119906 (1198971 119905) minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970119906 (1198971 119905)

=

8119890

1198970

int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (1198971 119905)

(8)

Similarly we can get the expression of Δ1198972

V

Δ1198971

V=

8119890

1198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (119909

1|=1198971 119905)]

Δ1198972

V=

8119890

1198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus (05119897

0minus 1198971) 119906 (1199092|=0 119905)]

(9)

In the static configuration the effective length of cablesegments 1 and 2 is given by the following equation [23]

119897119895

119890= int

119897119895

0

(

119889119904

119889119909119895

)

3

119889119909119895 (10)

where 119904 is the arch length of the cableBy integral operation the sagging expression for the first

cable segment can be written as follows as shown in Figure 2

119889lowast=

1198981198921198971

2 cos 120579lowast

8119867

=

1198981198921198971

2 cos (120579 minus 120572)8119867

(11)

Thus

1198971

119890= 1198970

[[

[

1205831+ 811989021205831

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

]]

]

1198972

119890= 1198970(1205832+ 81198902(1 minus 120583

1

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

))

(12)

where 1205831= 11989711198970is the relative position of transverse forces

and 1205832= 11989721198970= 1 minus 120583

1

From (9) and (12) the following formula is obtained

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ 1198970(05 minus 120583

1) 119906 (119909

1|=1198971 119905)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus 1198970(05 minus 120583

1) 119906 (1199092|=0 119905)]

(13)

Limited to the case when 119895 = 1 2 the configuration ofthe dynamic displacement can be written as follows usingseparation of variables

119906 (119909119895 119905) = 120593 (119909

119895) ei120596119905 (14)

4 Shock and Vibration

l0

x0

y

120572

120579

120579 120579lowast

Figure 2 Geometry of acted positions of transverse forces

where e(sdot) is the exponential function based on naturallogarithms 119894 = radicminus1 120596 is the angular frequency of vibrationand 120593(119909

119895) is the vibration mode function on segment 119895 By

substituting (14) into (13) the following formula is obtained

ℎ119895=ℎ119895ei120596119905 (15)

where

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

120593 (1199091) 1198891199091+ 1198970(05 minus 120583

1) 120593 (119909

1|=1198971)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

120593 (1199092) 1198891199092minus 1198970(05 minus 120583

1) 120593 (1199092|=0)]

(16)

It can be observed by comparing the above equationwith the equation given in literature [17 18] that whenthe effects of transverse forces are considered the formulafor calculating the coefficients of additional cable forces isrelevant to both the vibrational mode and the transverseforces

By substituting (14) into (1) the ordinary differentialequation for the vibrational mode function 120593(119909

119895) is obtained

as follows

120593119868119881(119909119895) minus

119867

119864119868

12059310158401015840(119909119895) minus

1198981205962

119864119868

120593 (119909119895) = minus

ℎ119895

8119890

1198641198681198970

(17)

Let

120585119895=

119909119895

1198970

120593 (120585119895) =

120593 (119909119895) sdot 119864119868

(1198981198921198974

0)

ℎ119895=

ℎ119895cos 120579119867

(18)

The dimensionless vibration equation for the two cablesegments is listed as follows

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = minus

ℎ119895 (19)

where 1205742 = 1198671198970

2119864119868 is the ratio of the axial force to bending

stiffness of the cable = 120596(1198972

0radic119864119868119898) = 120587

2(120596120596119861

1) =

120587120574(120596120596119878

1) is the dimensionless frequency of vibration 120596119904

1=

(1205871198970)radic119867119898 is the corresponding fundamental frequency of

a taut cable and 1205961198611= (12058721198970

2)radic119864119868119898 is the corresponding

fundamental frequency of a simple beamEquation (19) corresponds to the free vibration equation

of cable segment 119895 and its corresponding homogeneousequation can be written as

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = 0 (20)

The general solution of this ordinary differential equationis well solved in references [18 22] which can be written asthe following expression

120593lowast(120585119895) = 119860

119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895)

+ 119860119895

3cos (119902120585

119895) + 119860119895

4sin (119902120585

119895)

(21)

The special solution of (19) with the nonhomogeneousitem being constant can be written as

120593lowastlowast(120585119895) =

ℎ119895

2

(22)

Finally the complete solution of (19) can be written as

120593 (120585119895) = 120593lowast(120585119895) + 120593lowastlowast(120585119895)

= 119860119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895) + 119860119895

3cos (119902120585

119895)

+ 119860119895

4sin (119902120585

119895) +

ℎ119895

2

(23)

where

119901

119902 =

radicradic(

1205742

2

)

2

+ 2plusmn

1205742

2

=radicradic

(

1198671198972

0

2119864119868

)

2

+ 12059621198981198974

0

119864119868

plusmn

1198671198972

0

2119864119868

(24)

In addition it can be obtained from (24) that

119901119902 = = 120596

1198970

2

radic119864119868119898

1199012minus 1199022= 1205742=

1198671198970

2

119864119868

(25)

Equation (23) can be transformed to a matrix form asfollows

120593 (120585119895) = Φ (120585

119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

+

ℎ119895

2

(26)

Shock and Vibration 5

where

Φ (120585119895) = [eminus119901120585119895 eminus119901(120583119895minus120585119895) cos (119902120585

119895) sin (119902120585

119895)] 119895 = 1 2

(27)

This samemethod is adopted tomake (16) dimensionlessso that

ℎ1= 1205781sdot [int

1205831

0

120593 (1205851) 1198891205851+ (05 minus 120583

1) 120593 (1205851|=1205831)]

ℎ2= 1205782sdot [int

1205832

0

120593 (1205852) 1198891205852minus (05 minus 120583

1) 120593 (1205852|=0)]

(28)

In the above equation 120578119895indicates a parameter deter-

mined entirely by geometric parameters of the cable andtransverse force elements and is given by

120578119895= 64

1198601198970

3

119868119897119895

1198901198902 (29)

By substituting (23) into (28) subsequent arrangementand transposition a particular solution that is independentof the vibration mode equation is obtained and can besubsequently combined with the general solution to obtainthe particular solution of the vibration equation on cablesegment 119895 which is shown as follows

ℎ119895

2= B(119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(30)

where matrix B(119895) shows the impact of sagging on thedynamic system which can be calculated using the followingformula

B(119895) = 1198870

(119895)sdot

int

1205831

0

Φ (120585) 119889120585 + (05 minus 1205831)Φ (120585

1|=1205831) 119895 = 1

int

1205832

0

Φ (120585) 119889120585 minus (05 minus 1205831)Φ (1205852|=0) 119895 = 2

(31)

When the effects of sagging are considered the solutionfor cable segment 119895 can be given in the form of the generalsolution and the particular solution namely

120593 (120585119895) = (Φ (120585

119895) + B(119895)) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(32)

The amplitude of the additional cable force can beexpressed as

ℎ119895= 119867

2sec120579 sdot B(119895) sdot 1198601

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(33)

3 Dimensionless Vibrational Mode Function

The undetermined coefficients in the solutions of the vibra-tional mode function solution for two cable segments givenby formula (32) can be determined through the boundaryconditions

1l 2l

Maei120596t Mcei120596t Mce

i120596tVce

i120596t

Vaei120596t Vbei120596t

Vcei120596t

Vcei120596tVcei120596t

Mbei120596t

120572aei120596t 120572cei120596t 120572cei120596t120572cei120596t

120572bei120596t120579aei120596t

120579bei120596t

120579cei120596t 120579cei120596t

TTTT

Figure 3 A mechanical model of a discretized cable dampersystem

As shown in Figure 3 the cable system is divided intotwo independent segments and a transverse force elementThe dynamic nodal force and dynamic nodal displacementsact on the boundaries of these models After discretizationthe nodal forces for each element include the amplitudeof the dynamic nodal shear forces 119881

119886 119881119888 and 119881

119887and the

amplitude of dynamic nodal bending moments 119872119886 119872119888

and119872119887 The nodal displacements include the amplitudes of

dynamic displacement 120572119886 120572119888 and 120572

119887and the dynamic swing

amplitudes 120579119886 120579119888 and 120579

119887 The time-varying part of these

quantities is represented by a complex exponent ei120596119905 wherei = radicminus1 is the complex unit

The boundary conditions of the displacement of the twocable segments are given by the following formula

120572119886=

1198981198921198974

0

119864119868

120593 (1205851|=0)

1205791198861198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=0)

120572119888=

1198981198921198974

0

119864119868

120593 (1205851|=1205831)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=1205831)

(34)

120572119888=

1198981198921198974

0

119864119868

120593 (1205852|=0)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=0)

120572119887=

1198981198921198974

0

119864119868

120593 (1205852|=1205832)

1205791198871198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=1205832)

(35)

For cable segments 119895 = 1 2 each of the nodal displace-ments 120572

119886 1205791198861198970 120572119888 1205791198881198970119879 and 120572

119888 1205791198881198970 120572119887 1205791198871198970119879 is used as

boundary condition respectively to obtain the expressions

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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2 Shock and Vibration

hinges which took sagging into consideration and analyzedthe features of its elements [12 13] Sarkar and Manoharproposed a calculation scheme for the dynamic stiffnessmatrix of cables using finite element analysis [14] In 2001Kim and Chang gave an approximate analytical expressionof the dynamic stiffness matrix of an inclined cable with acatenary configuration [15] which surpassed the limitationsof the parabolic static configuration

Each of these studies on dynamic stiffness consideredvarious factors based on the taut string theory while theterms corresponding to the bending stiffness of the cableswere all neglected in the passive vibration control equationsThe actual cable structures are usually inclined cables witheach end clamped and with relatively high bending stiffness[16 17] With increased length the effect of sag cannot beneglected [18 19] A calculation algorithm of the dynamicstiffness of the cables with all of these factors taken intoaccount would be very valuable to engineering practicesFurthermore dynamic stiffness is used to describe thedynamical behavior at the ends of cables and thereforecannot be used in the case of a cable acted upon by transverseforces However in engineering practices a majority of cablesystems are acted upon by transverse forces and thereforethe study of transverse dynamic stiffness of cables is crucialand will contribute substantially to studies involving cabledamping systems [20 21]

In this study a cable is modeled as being divided intovarious segments each acted upon by a unique transverseforce while factors including sagging bending stiffnessclamped boundary conditions and inclination angle are allconsideredThe dynamic stiffness method is used to establishan equation of motion and its general solution correspondsto a cable system acted upon by transverse forces Thedynamic displacement of each segment is used to characterizethe dimensionless vibration mode function Eventually ananalytical expression for the transverse dynamic stiffnessmatrix of a cable is given in a clear simple form The resultsof the study lay a foundation for further research in systemvibration

2 General Solution forGoverning Equation of Cable

The focus of this study is on a commonly observed situationin engineering namely when one end of a cable is acted uponby transverse forces As shown in Figure 1 due to the presenceof transverse forces the cable system can be discretized intocable segment 1 (from end a to end c) cable segment 2(from end c to end b) and transverse force elements (suchas dampers)

As shown in Figure 1 by assuming that transverse forcesdonot affect the static configuration of the cable the two cablesegments can be described by the same quadratic parabolicfunction 119910(119909

0) when the cable vibrates segments 1 and 2

follow different configurations namely 119906(1199091 119905) and 119906(119909

2 119905)

When considering the effects of sagging bending stiffnessclamped boundary conditions and inclination angle andignoring the internal damping and shear forces following

H

l0

l1

l2

x0

x1

x2

d

y(x0) = y(x2 + l1)

120579

y(x0) 120597u(x2 t)

120597x2dx2

dx2

a

b

c

d(Δl) asymp ( 120597u(x2 t)120597x2minusdy(x0)

dx0) middot dy(x0)

dx0dx2

u(x2 t)

u(x2 t)

u(x1 t)

Figure 1 Schematic diagram of calculation of additional dynamicstress

governing equation [18 22] can be used to describe the freevibration of the two segments

119864119868

1205974119906

1205971199094

119895

minus 119867

1205972119906

1205971199092

119895

minus ℎ119895

1198892119910

11988911990902+ 119898

1205972119906

1205971199052= 0 (1)

where 119895 = 1 2 119864119868 is the flexural rigidity 119867 is the initialtension force 119898 is the mass per unit length 119910 is theconfiguration of the cable under static equilibrium (assumingthat dampers have no impact on the static configuration)119909119895is the coordinate of each segment and ℎ

119895represents the

additional segmented force which is an additional cabletension generated due to the difference from the staticconfiguration during vibration and is equal to the product ofthe additional strain 120576

119895

V(119905) generated during difference from

the static equilibrium position and the axial stiffnessThat is

ℎ119895= 119864119860120576

119895

V(119905) (2)

The additional dynamic strain 120576119895

V(119905) can be calculated

using the ratio between the change in cable lengthΔ119897119895

V duringvibration and the static cable length 119897

119895

119890 Consider

120576119895

V(119905) =

Δ119897119895

V

119897119895

119890 (3)

In (2) and (3) the superscript V represents the dynamicconfiguration and 119890 represents static equilibrium A similarrule is followed below

As shown in Figure 1 in bridge projects the anglebetween the tangential direction and the chord of a cable isvery small for both the static and dynamic configurationsTherefore when ignoring the higher order terms of the staticand dynamic configurations the amount of elongation 119889119909

119895of

Shock and Vibration 3

a differential segment during vibration relative to that of thestatic configuration 119889(Δ119897

119895

V) can be approximated by

119889 (Δ119897119895

V) asymp (

120597119906 (119909119895 119905)

120597119909119895

minus

119889119910 (1199090)

1198891199090

) sdot

119889119910 (1199090)

1198891199090

119889119909119895

asymp

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895

(4)

The total elongation of segment 119895 can be written as

Δ119897119895

V= int

119897119895

0

119889 (Δ119897119895

V) = int

119897119895

0

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895 (5)

With the assumption of a parabolic static equilibriumone can use the following expression for the configuration

119910 (1199090) = minus

4119890

1198970

1199090(1199090minus 1198970)

= minus

4119890

1198970

(1199092+ 1198971) (1199092minus 1198972)

= minus

4119890

1198970

1199091(1199091minus 1198970)

(6)

where 119890 = 1198891198970= 119898119892119897

0cos 1205798119867 is the sag-span ratio and

119889 = 1198981198921198970

2 cos 1205798119867Starting from (6) and noting the relationship of 119909

0= 1199091=

1199092+1198971 we can easily get the coordinate derivative of the static

configuration function 119910(1199090) as follows

119889119910 (1199090)

1198891199090

= minus

4119890

1198970

[21199090minus 1198970] = minus

4119890

1198970

[21199091minus 1198970]

= minus

4119890

1198970

[21199092+ (1198971minus 1198972)]

(7)

And then by substituting it into (5) we have

Δ1198971

V= int

1198971

0

119889 (Δ1198971

V) = int

1198971

0

120597119906 (1199091 119905)

1205971199091

119889119910 (1199090)

1198891199090

1198891199091

= minus

4119890

1198970

int

1198971

0

120597119906 (1199091 119905)

1205971199091

[21199091minus 1198970] 1198891199091

= minus

4119890

1198970

2int

1198971

0

120597119906 (1199091 119905)

1205971199091

11990911198891199091minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2int

1198971

0

120597

1205971199091

[119906 (1199091 119905) 1199091] 1198891199091

minus2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2[119906 (1199091 119905) 1199091]1198971

0minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1

minus 1198970119906(1199091 119905)1198971

0

= minus

4119890

1198970

21198971119906 (1198971 119905) minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970119906 (1198971 119905)

=

8119890

1198970

int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (1198971 119905)

(8)

Similarly we can get the expression of Δ1198972

V

Δ1198971

V=

8119890

1198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (119909

1|=1198971 119905)]

Δ1198972

V=

8119890

1198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus (05119897

0minus 1198971) 119906 (1199092|=0 119905)]

(9)

In the static configuration the effective length of cablesegments 1 and 2 is given by the following equation [23]

119897119895

119890= int

119897119895

0

(

119889119904

119889119909119895

)

3

119889119909119895 (10)

where 119904 is the arch length of the cableBy integral operation the sagging expression for the first

cable segment can be written as follows as shown in Figure 2

119889lowast=

1198981198921198971

2 cos 120579lowast

8119867

=

1198981198921198971

2 cos (120579 minus 120572)8119867

(11)

Thus

1198971

119890= 1198970

[[

[

1205831+ 811989021205831

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

]]

]

1198972

119890= 1198970(1205832+ 81198902(1 minus 120583

1

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

))

(12)

where 1205831= 11989711198970is the relative position of transverse forces

and 1205832= 11989721198970= 1 minus 120583

1

From (9) and (12) the following formula is obtained

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ 1198970(05 minus 120583

1) 119906 (119909

1|=1198971 119905)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus 1198970(05 minus 120583

1) 119906 (1199092|=0 119905)]

(13)

Limited to the case when 119895 = 1 2 the configuration ofthe dynamic displacement can be written as follows usingseparation of variables

119906 (119909119895 119905) = 120593 (119909

119895) ei120596119905 (14)

4 Shock and Vibration

l0

x0

y

120572

120579

120579 120579lowast

Figure 2 Geometry of acted positions of transverse forces

where e(sdot) is the exponential function based on naturallogarithms 119894 = radicminus1 120596 is the angular frequency of vibrationand 120593(119909

119895) is the vibration mode function on segment 119895 By

substituting (14) into (13) the following formula is obtained

ℎ119895=ℎ119895ei120596119905 (15)

where

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

120593 (1199091) 1198891199091+ 1198970(05 minus 120583

1) 120593 (119909

1|=1198971)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

120593 (1199092) 1198891199092minus 1198970(05 minus 120583

1) 120593 (1199092|=0)]

(16)

It can be observed by comparing the above equationwith the equation given in literature [17 18] that whenthe effects of transverse forces are considered the formulafor calculating the coefficients of additional cable forces isrelevant to both the vibrational mode and the transverseforces

By substituting (14) into (1) the ordinary differentialequation for the vibrational mode function 120593(119909

119895) is obtained

as follows

120593119868119881(119909119895) minus

119867

119864119868

12059310158401015840(119909119895) minus

1198981205962

119864119868

120593 (119909119895) = minus

ℎ119895

8119890

1198641198681198970

(17)

Let

120585119895=

119909119895

1198970

120593 (120585119895) =

120593 (119909119895) sdot 119864119868

(1198981198921198974

0)

ℎ119895=

ℎ119895cos 120579119867

(18)

The dimensionless vibration equation for the two cablesegments is listed as follows

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = minus

ℎ119895 (19)

where 1205742 = 1198671198970

2119864119868 is the ratio of the axial force to bending

stiffness of the cable = 120596(1198972

0radic119864119868119898) = 120587

2(120596120596119861

1) =

120587120574(120596120596119878

1) is the dimensionless frequency of vibration 120596119904

1=

(1205871198970)radic119867119898 is the corresponding fundamental frequency of

a taut cable and 1205961198611= (12058721198970

2)radic119864119868119898 is the corresponding

fundamental frequency of a simple beamEquation (19) corresponds to the free vibration equation

of cable segment 119895 and its corresponding homogeneousequation can be written as

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = 0 (20)

The general solution of this ordinary differential equationis well solved in references [18 22] which can be written asthe following expression

120593lowast(120585119895) = 119860

119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895)

+ 119860119895

3cos (119902120585

119895) + 119860119895

4sin (119902120585

119895)

(21)

The special solution of (19) with the nonhomogeneousitem being constant can be written as

120593lowastlowast(120585119895) =

ℎ119895

2

(22)

Finally the complete solution of (19) can be written as

120593 (120585119895) = 120593lowast(120585119895) + 120593lowastlowast(120585119895)

= 119860119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895) + 119860119895

3cos (119902120585

119895)

+ 119860119895

4sin (119902120585

119895) +

ℎ119895

2

(23)

where

119901

119902 =

radicradic(

1205742

2

)

2

+ 2plusmn

1205742

2

=radicradic

(

1198671198972

0

2119864119868

)

2

+ 12059621198981198974

0

119864119868

plusmn

1198671198972

0

2119864119868

(24)

In addition it can be obtained from (24) that

119901119902 = = 120596

1198970

2

radic119864119868119898

1199012minus 1199022= 1205742=

1198671198970

2

119864119868

(25)

Equation (23) can be transformed to a matrix form asfollows

120593 (120585119895) = Φ (120585

119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

+

ℎ119895

2

(26)

Shock and Vibration 5

where

Φ (120585119895) = [eminus119901120585119895 eminus119901(120583119895minus120585119895) cos (119902120585

119895) sin (119902120585

119895)] 119895 = 1 2

(27)

This samemethod is adopted tomake (16) dimensionlessso that

ℎ1= 1205781sdot [int

1205831

0

120593 (1205851) 1198891205851+ (05 minus 120583

1) 120593 (1205851|=1205831)]

ℎ2= 1205782sdot [int

1205832

0

120593 (1205852) 1198891205852minus (05 minus 120583

1) 120593 (1205852|=0)]

(28)

In the above equation 120578119895indicates a parameter deter-

mined entirely by geometric parameters of the cable andtransverse force elements and is given by

120578119895= 64

1198601198970

3

119868119897119895

1198901198902 (29)

By substituting (23) into (28) subsequent arrangementand transposition a particular solution that is independentof the vibration mode equation is obtained and can besubsequently combined with the general solution to obtainthe particular solution of the vibration equation on cablesegment 119895 which is shown as follows

ℎ119895

2= B(119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(30)

where matrix B(119895) shows the impact of sagging on thedynamic system which can be calculated using the followingformula

B(119895) = 1198870

(119895)sdot

int

1205831

0

Φ (120585) 119889120585 + (05 minus 1205831)Φ (120585

1|=1205831) 119895 = 1

int

1205832

0

Φ (120585) 119889120585 minus (05 minus 1205831)Φ (1205852|=0) 119895 = 2

(31)

When the effects of sagging are considered the solutionfor cable segment 119895 can be given in the form of the generalsolution and the particular solution namely

120593 (120585119895) = (Φ (120585

119895) + B(119895)) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(32)

The amplitude of the additional cable force can beexpressed as

ℎ119895= 119867

2sec120579 sdot B(119895) sdot 1198601

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(33)

3 Dimensionless Vibrational Mode Function

The undetermined coefficients in the solutions of the vibra-tional mode function solution for two cable segments givenby formula (32) can be determined through the boundaryconditions

1l 2l

Maei120596t Mcei120596t Mce

i120596tVce

i120596t

Vaei120596t Vbei120596t

Vcei120596t

Vcei120596tVcei120596t

Mbei120596t

120572aei120596t 120572cei120596t 120572cei120596t120572cei120596t

120572bei120596t120579aei120596t

120579bei120596t

120579cei120596t 120579cei120596t

TTTT

Figure 3 A mechanical model of a discretized cable dampersystem

As shown in Figure 3 the cable system is divided intotwo independent segments and a transverse force elementThe dynamic nodal force and dynamic nodal displacementsact on the boundaries of these models After discretizationthe nodal forces for each element include the amplitudeof the dynamic nodal shear forces 119881

119886 119881119888 and 119881

119887and the

amplitude of dynamic nodal bending moments 119872119886 119872119888

and119872119887 The nodal displacements include the amplitudes of

dynamic displacement 120572119886 120572119888 and 120572

119887and the dynamic swing

amplitudes 120579119886 120579119888 and 120579

119887 The time-varying part of these

quantities is represented by a complex exponent ei120596119905 wherei = radicminus1 is the complex unit

The boundary conditions of the displacement of the twocable segments are given by the following formula

120572119886=

1198981198921198974

0

119864119868

120593 (1205851|=0)

1205791198861198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=0)

120572119888=

1198981198921198974

0

119864119868

120593 (1205851|=1205831)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=1205831)

(34)

120572119888=

1198981198921198974

0

119864119868

120593 (1205852|=0)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=0)

120572119887=

1198981198921198974

0

119864119868

120593 (1205852|=1205832)

1205791198871198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=1205832)

(35)

For cable segments 119895 = 1 2 each of the nodal displace-ments 120572

119886 1205791198861198970 120572119888 1205791198881198970119879 and 120572

119888 1205791198881198970 120572119887 1205791198871198970119879 is used as

boundary condition respectively to obtain the expressions

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

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Shock and Vibration 3

a differential segment during vibration relative to that of thestatic configuration 119889(Δ119897

119895

V) can be approximated by

119889 (Δ119897119895

V) asymp (

120597119906 (119909119895 119905)

120597119909119895

minus

119889119910 (1199090)

1198891199090

) sdot

119889119910 (1199090)

1198891199090

119889119909119895

asymp

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895

(4)

The total elongation of segment 119895 can be written as

Δ119897119895

V= int

119897119895

0

119889 (Δ119897119895

V) = int

119897119895

0

120597119906 (119909119895 119905)

120597119909119895

119889119910 (1199090)

1198891199090

119889119909119895 (5)

With the assumption of a parabolic static equilibriumone can use the following expression for the configuration

119910 (1199090) = minus

4119890

1198970

1199090(1199090minus 1198970)

= minus

4119890

1198970

(1199092+ 1198971) (1199092minus 1198972)

= minus

4119890

1198970

1199091(1199091minus 1198970)

(6)

where 119890 = 1198891198970= 119898119892119897

0cos 1205798119867 is the sag-span ratio and

119889 = 1198981198921198970

2 cos 1205798119867Starting from (6) and noting the relationship of 119909

0= 1199091=

1199092+1198971 we can easily get the coordinate derivative of the static

configuration function 119910(1199090) as follows

119889119910 (1199090)

1198891199090

= minus

4119890

1198970

[21199090minus 1198970] = minus

4119890

1198970

[21199091minus 1198970]

= minus

4119890

1198970

[21199092+ (1198971minus 1198972)]

(7)

And then by substituting it into (5) we have

Δ1198971

V= int

1198971

0

119889 (Δ1198971

V) = int

1198971

0

120597119906 (1199091 119905)

1205971199091

119889119910 (1199090)

1198891199090

1198891199091

= minus

4119890

1198970

int

1198971

0

120597119906 (1199091 119905)

1205971199091

[21199091minus 1198970] 1198891199091

= minus

4119890

1198970

2int

1198971

0

120597119906 (1199091 119905)

1205971199091

11990911198891199091minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2int

1198971

0

120597

1205971199091

[119906 (1199091 119905) 1199091] 1198891199091

minus2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970int

1198971

0

120597119906 (1199091 119905)

1205971199091

1198891199091

= minus

4119890

1198970

2[119906 (1199091 119905) 1199091]1198971

0minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1

minus 1198970119906(1199091 119905)1198971

0

= minus

4119890

1198970

21198971119906 (1198971 119905) minus 2int

1198971

0

119906 (1199091 119905) 119889119909

1minus 1198970119906 (1198971 119905)

=

8119890

1198970

int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (1198971 119905)

(8)

Similarly we can get the expression of Δ1198972

V

Δ1198971

V=

8119890

1198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ (05119897

0minus 1198971) 119906 (119909

1|=1198971 119905)]

Δ1198972

V=

8119890

1198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus (05119897

0minus 1198971) 119906 (1199092|=0 119905)]

(9)

In the static configuration the effective length of cablesegments 1 and 2 is given by the following equation [23]

119897119895

119890= int

119897119895

0

(

119889119904

119889119909119895

)

3

119889119909119895 (10)

where 119904 is the arch length of the cableBy integral operation the sagging expression for the first

cable segment can be written as follows as shown in Figure 2

119889lowast=

1198981198921198971

2 cos 120579lowast

8119867

=

1198981198921198971

2 cos (120579 minus 120572)8119867

(11)

Thus

1198971

119890= 1198970

[[

[

1205831+ 811989021205831

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

]]

]

1198972

119890= 1198970(1205832+ 81198902(1 minus 120583

1

3(

1 + 41198901205832tan 120579

radic1 + 16119890212058322

)

2

))

(12)

where 1205831= 11989711198970is the relative position of transverse forces

and 1205832= 11989721198970= 1 minus 120583

1

From (9) and (12) the following formula is obtained

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

119906 (1199091 119905) 119889119909

1+ 1198970(05 minus 120583

1) 119906 (119909

1|=1198971 119905)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

119906 (1199092 119905) 119889119909

2minus 1198970(05 minus 120583

1) 119906 (1199092|=0 119905)]

(13)

Limited to the case when 119895 = 1 2 the configuration ofthe dynamic displacement can be written as follows usingseparation of variables

119906 (119909119895 119905) = 120593 (119909

119895) ei120596119905 (14)

4 Shock and Vibration

l0

x0

y

120572

120579

120579 120579lowast

Figure 2 Geometry of acted positions of transverse forces

where e(sdot) is the exponential function based on naturallogarithms 119894 = radicminus1 120596 is the angular frequency of vibrationand 120593(119909

119895) is the vibration mode function on segment 119895 By

substituting (14) into (13) the following formula is obtained

ℎ119895=ℎ119895ei120596119905 (15)

where

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

120593 (1199091) 1198891199091+ 1198970(05 minus 120583

1) 120593 (119909

1|=1198971)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

120593 (1199092) 1198891199092minus 1198970(05 minus 120583

1) 120593 (1199092|=0)]

(16)

It can be observed by comparing the above equationwith the equation given in literature [17 18] that whenthe effects of transverse forces are considered the formulafor calculating the coefficients of additional cable forces isrelevant to both the vibrational mode and the transverseforces

By substituting (14) into (1) the ordinary differentialequation for the vibrational mode function 120593(119909

119895) is obtained

as follows

120593119868119881(119909119895) minus

119867

119864119868

12059310158401015840(119909119895) minus

1198981205962

119864119868

120593 (119909119895) = minus

ℎ119895

8119890

1198641198681198970

(17)

Let

120585119895=

119909119895

1198970

120593 (120585119895) =

120593 (119909119895) sdot 119864119868

(1198981198921198974

0)

ℎ119895=

ℎ119895cos 120579119867

(18)

The dimensionless vibration equation for the two cablesegments is listed as follows

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = minus

ℎ119895 (19)

where 1205742 = 1198671198970

2119864119868 is the ratio of the axial force to bending

stiffness of the cable = 120596(1198972

0radic119864119868119898) = 120587

2(120596120596119861

1) =

120587120574(120596120596119878

1) is the dimensionless frequency of vibration 120596119904

1=

(1205871198970)radic119867119898 is the corresponding fundamental frequency of

a taut cable and 1205961198611= (12058721198970

2)radic119864119868119898 is the corresponding

fundamental frequency of a simple beamEquation (19) corresponds to the free vibration equation

of cable segment 119895 and its corresponding homogeneousequation can be written as

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = 0 (20)

The general solution of this ordinary differential equationis well solved in references [18 22] which can be written asthe following expression

120593lowast(120585119895) = 119860

119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895)

+ 119860119895

3cos (119902120585

119895) + 119860119895

4sin (119902120585

119895)

(21)

The special solution of (19) with the nonhomogeneousitem being constant can be written as

120593lowastlowast(120585119895) =

ℎ119895

2

(22)

Finally the complete solution of (19) can be written as

120593 (120585119895) = 120593lowast(120585119895) + 120593lowastlowast(120585119895)

= 119860119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895) + 119860119895

3cos (119902120585

119895)

+ 119860119895

4sin (119902120585

119895) +

ℎ119895

2

(23)

where

119901

119902 =

radicradic(

1205742

2

)

2

+ 2plusmn

1205742

2

=radicradic

(

1198671198972

0

2119864119868

)

2

+ 12059621198981198974

0

119864119868

plusmn

1198671198972

0

2119864119868

(24)

In addition it can be obtained from (24) that

119901119902 = = 120596

1198970

2

radic119864119868119898

1199012minus 1199022= 1205742=

1198671198970

2

119864119868

(25)

Equation (23) can be transformed to a matrix form asfollows

120593 (120585119895) = Φ (120585

119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

+

ℎ119895

2

(26)

Shock and Vibration 5

where

Φ (120585119895) = [eminus119901120585119895 eminus119901(120583119895minus120585119895) cos (119902120585

119895) sin (119902120585

119895)] 119895 = 1 2

(27)

This samemethod is adopted tomake (16) dimensionlessso that

ℎ1= 1205781sdot [int

1205831

0

120593 (1205851) 1198891205851+ (05 minus 120583

1) 120593 (1205851|=1205831)]

ℎ2= 1205782sdot [int

1205832

0

120593 (1205852) 1198891205852minus (05 minus 120583

1) 120593 (1205852|=0)]

(28)

In the above equation 120578119895indicates a parameter deter-

mined entirely by geometric parameters of the cable andtransverse force elements and is given by

120578119895= 64

1198601198970

3

119868119897119895

1198901198902 (29)

By substituting (23) into (28) subsequent arrangementand transposition a particular solution that is independentof the vibration mode equation is obtained and can besubsequently combined with the general solution to obtainthe particular solution of the vibration equation on cablesegment 119895 which is shown as follows

ℎ119895

2= B(119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(30)

where matrix B(119895) shows the impact of sagging on thedynamic system which can be calculated using the followingformula

B(119895) = 1198870

(119895)sdot

int

1205831

0

Φ (120585) 119889120585 + (05 minus 1205831)Φ (120585

1|=1205831) 119895 = 1

int

1205832

0

Φ (120585) 119889120585 minus (05 minus 1205831)Φ (1205852|=0) 119895 = 2

(31)

When the effects of sagging are considered the solutionfor cable segment 119895 can be given in the form of the generalsolution and the particular solution namely

120593 (120585119895) = (Φ (120585

119895) + B(119895)) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(32)

The amplitude of the additional cable force can beexpressed as

ℎ119895= 119867

2sec120579 sdot B(119895) sdot 1198601

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(33)

3 Dimensionless Vibrational Mode Function

The undetermined coefficients in the solutions of the vibra-tional mode function solution for two cable segments givenby formula (32) can be determined through the boundaryconditions

1l 2l

Maei120596t Mcei120596t Mce

i120596tVce

i120596t

Vaei120596t Vbei120596t

Vcei120596t

Vcei120596tVcei120596t

Mbei120596t

120572aei120596t 120572cei120596t 120572cei120596t120572cei120596t

120572bei120596t120579aei120596t

120579bei120596t

120579cei120596t 120579cei120596t

TTTT

Figure 3 A mechanical model of a discretized cable dampersystem

As shown in Figure 3 the cable system is divided intotwo independent segments and a transverse force elementThe dynamic nodal force and dynamic nodal displacementsact on the boundaries of these models After discretizationthe nodal forces for each element include the amplitudeof the dynamic nodal shear forces 119881

119886 119881119888 and 119881

119887and the

amplitude of dynamic nodal bending moments 119872119886 119872119888

and119872119887 The nodal displacements include the amplitudes of

dynamic displacement 120572119886 120572119888 and 120572

119887and the dynamic swing

amplitudes 120579119886 120579119888 and 120579

119887 The time-varying part of these

quantities is represented by a complex exponent ei120596119905 wherei = radicminus1 is the complex unit

The boundary conditions of the displacement of the twocable segments are given by the following formula

120572119886=

1198981198921198974

0

119864119868

120593 (1205851|=0)

1205791198861198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=0)

120572119888=

1198981198921198974

0

119864119868

120593 (1205851|=1205831)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=1205831)

(34)

120572119888=

1198981198921198974

0

119864119868

120593 (1205852|=0)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=0)

120572119887=

1198981198921198974

0

119864119868

120593 (1205852|=1205832)

1205791198871198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=1205832)

(35)

For cable segments 119895 = 1 2 each of the nodal displace-ments 120572

119886 1205791198861198970 120572119888 1205791198881198970119879 and 120572

119888 1205791198881198970 120572119887 1205791198871198970119879 is used as

boundary condition respectively to obtain the expressions

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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4 Shock and Vibration

l0

x0

y

120572

120579

120579 120579lowast

Figure 2 Geometry of acted positions of transverse forces

where e(sdot) is the exponential function based on naturallogarithms 119894 = radicminus1 120596 is the angular frequency of vibrationand 120593(119909

119895) is the vibration mode function on segment 119895 By

substituting (14) into (13) the following formula is obtained

ℎ119895=ℎ119895ei120596119905 (15)

where

ℎ1=

8119864119860119890

1198971

1198901198970

[int

1198971

0

120593 (1199091) 1198891199091+ 1198970(05 minus 120583

1) 120593 (119909

1|=1198971)]

ℎ2=

8119864119860119890

1198972

1198901198970

[int

1198972

0

120593 (1199092) 1198891199092minus 1198970(05 minus 120583

1) 120593 (1199092|=0)]

(16)

It can be observed by comparing the above equationwith the equation given in literature [17 18] that whenthe effects of transverse forces are considered the formulafor calculating the coefficients of additional cable forces isrelevant to both the vibrational mode and the transverseforces

By substituting (14) into (1) the ordinary differentialequation for the vibrational mode function 120593(119909

119895) is obtained

as follows

120593119868119881(119909119895) minus

119867

119864119868

12059310158401015840(119909119895) minus

1198981205962

119864119868

120593 (119909119895) = minus

ℎ119895

8119890

1198641198681198970

(17)

Let

120585119895=

119909119895

1198970

120593 (120585119895) =

120593 (119909119895) sdot 119864119868

(1198981198921198974

0)

ℎ119895=

ℎ119895cos 120579119867

(18)

The dimensionless vibration equation for the two cablesegments is listed as follows

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = minus

ℎ119895 (19)

where 1205742 = 1198671198970

2119864119868 is the ratio of the axial force to bending

stiffness of the cable = 120596(1198972

0radic119864119868119898) = 120587

2(120596120596119861

1) =

120587120574(120596120596119878

1) is the dimensionless frequency of vibration 120596119904

1=

(1205871198970)radic119867119898 is the corresponding fundamental frequency of

a taut cable and 1205961198611= (12058721198970

2)radic119864119868119898 is the corresponding

fundamental frequency of a simple beamEquation (19) corresponds to the free vibration equation

of cable segment 119895 and its corresponding homogeneousequation can be written as

120593119868119881(120585119895) minus 120574212059310158401015840(120585119895) minus 2120593 (120585119895) = 0 (20)

The general solution of this ordinary differential equationis well solved in references [18 22] which can be written asthe following expression

120593lowast(120585119895) = 119860

119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895)

+ 119860119895

3cos (119902120585

119895) + 119860119895

4sin (119902120585

119895)

(21)

The special solution of (19) with the nonhomogeneousitem being constant can be written as

120593lowastlowast(120585119895) =

ℎ119895

2

(22)

Finally the complete solution of (19) can be written as

120593 (120585119895) = 120593lowast(120585119895) + 120593lowastlowast(120585119895)

= 119860119895

1eminus119901120585119895 + 119860119895

2eminus119901(120583119895minus120585119895) + 119860119895

3cos (119902120585

119895)

+ 119860119895

4sin (119902120585

119895) +

ℎ119895

2

(23)

where

119901

119902 =

radicradic(

1205742

2

)

2

+ 2plusmn

1205742

2

=radicradic

(

1198671198972

0

2119864119868

)

2

+ 12059621198981198974

0

119864119868

plusmn

1198671198972

0

2119864119868

(24)

In addition it can be obtained from (24) that

119901119902 = = 120596

1198970

2

radic119864119868119898

1199012minus 1199022= 1205742=

1198671198970

2

119864119868

(25)

Equation (23) can be transformed to a matrix form asfollows

120593 (120585119895) = Φ (120585

119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

+

ℎ119895

2

(26)

Shock and Vibration 5

where

Φ (120585119895) = [eminus119901120585119895 eminus119901(120583119895minus120585119895) cos (119902120585

119895) sin (119902120585

119895)] 119895 = 1 2

(27)

This samemethod is adopted tomake (16) dimensionlessso that

ℎ1= 1205781sdot [int

1205831

0

120593 (1205851) 1198891205851+ (05 minus 120583

1) 120593 (1205851|=1205831)]

ℎ2= 1205782sdot [int

1205832

0

120593 (1205852) 1198891205852minus (05 minus 120583

1) 120593 (1205852|=0)]

(28)

In the above equation 120578119895indicates a parameter deter-

mined entirely by geometric parameters of the cable andtransverse force elements and is given by

120578119895= 64

1198601198970

3

119868119897119895

1198901198902 (29)

By substituting (23) into (28) subsequent arrangementand transposition a particular solution that is independentof the vibration mode equation is obtained and can besubsequently combined with the general solution to obtainthe particular solution of the vibration equation on cablesegment 119895 which is shown as follows

ℎ119895

2= B(119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(30)

where matrix B(119895) shows the impact of sagging on thedynamic system which can be calculated using the followingformula

B(119895) = 1198870

(119895)sdot

int

1205831

0

Φ (120585) 119889120585 + (05 minus 1205831)Φ (120585

1|=1205831) 119895 = 1

int

1205832

0

Φ (120585) 119889120585 minus (05 minus 1205831)Φ (1205852|=0) 119895 = 2

(31)

When the effects of sagging are considered the solutionfor cable segment 119895 can be given in the form of the generalsolution and the particular solution namely

120593 (120585119895) = (Φ (120585

119895) + B(119895)) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(32)

The amplitude of the additional cable force can beexpressed as

ℎ119895= 119867

2sec120579 sdot B(119895) sdot 1198601

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(33)

3 Dimensionless Vibrational Mode Function

The undetermined coefficients in the solutions of the vibra-tional mode function solution for two cable segments givenby formula (32) can be determined through the boundaryconditions

1l 2l

Maei120596t Mcei120596t Mce

i120596tVce

i120596t

Vaei120596t Vbei120596t

Vcei120596t

Vcei120596tVcei120596t

Mbei120596t

120572aei120596t 120572cei120596t 120572cei120596t120572cei120596t

120572bei120596t120579aei120596t

120579bei120596t

120579cei120596t 120579cei120596t

TTTT

Figure 3 A mechanical model of a discretized cable dampersystem

As shown in Figure 3 the cable system is divided intotwo independent segments and a transverse force elementThe dynamic nodal force and dynamic nodal displacementsact on the boundaries of these models After discretizationthe nodal forces for each element include the amplitudeof the dynamic nodal shear forces 119881

119886 119881119888 and 119881

119887and the

amplitude of dynamic nodal bending moments 119872119886 119872119888

and119872119887 The nodal displacements include the amplitudes of

dynamic displacement 120572119886 120572119888 and 120572

119887and the dynamic swing

amplitudes 120579119886 120579119888 and 120579

119887 The time-varying part of these

quantities is represented by a complex exponent ei120596119905 wherei = radicminus1 is the complex unit

The boundary conditions of the displacement of the twocable segments are given by the following formula

120572119886=

1198981198921198974

0

119864119868

120593 (1205851|=0)

1205791198861198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=0)

120572119888=

1198981198921198974

0

119864119868

120593 (1205851|=1205831)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=1205831)

(34)

120572119888=

1198981198921198974

0

119864119868

120593 (1205852|=0)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=0)

120572119887=

1198981198921198974

0

119864119868

120593 (1205852|=1205832)

1205791198871198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=1205832)

(35)

For cable segments 119895 = 1 2 each of the nodal displace-ments 120572

119886 1205791198861198970 120572119888 1205791198881198970119879 and 120572

119888 1205791198881198970 120572119887 1205791198871198970119879 is used as

boundary condition respectively to obtain the expressions

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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Shock and Vibration 5

where

Φ (120585119895) = [eminus119901120585119895 eminus119901(120583119895minus120585119895) cos (119902120585

119895) sin (119902120585

119895)] 119895 = 1 2

(27)

This samemethod is adopted tomake (16) dimensionlessso that

ℎ1= 1205781sdot [int

1205831

0

120593 (1205851) 1198891205851+ (05 minus 120583

1) 120593 (1205851|=1205831)]

ℎ2= 1205782sdot [int

1205832

0

120593 (1205852) 1198891205852minus (05 minus 120583

1) 120593 (1205852|=0)]

(28)

In the above equation 120578119895indicates a parameter deter-

mined entirely by geometric parameters of the cable andtransverse force elements and is given by

120578119895= 64

1198601198970

3

119868119897119895

1198901198902 (29)

By substituting (23) into (28) subsequent arrangementand transposition a particular solution that is independentof the vibration mode equation is obtained and can besubsequently combined with the general solution to obtainthe particular solution of the vibration equation on cablesegment 119895 which is shown as follows

ℎ119895

2= B(119895) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(30)

where matrix B(119895) shows the impact of sagging on thedynamic system which can be calculated using the followingformula

B(119895) = 1198870

(119895)sdot

int

1205831

0

Φ (120585) 119889120585 + (05 minus 1205831)Φ (120585

1|=1205831) 119895 = 1

int

1205832

0

Φ (120585) 119889120585 minus (05 minus 1205831)Φ (1205852|=0) 119895 = 2

(31)

When the effects of sagging are considered the solutionfor cable segment 119895 can be given in the form of the generalsolution and the particular solution namely

120593 (120585119895) = (Φ (120585

119895) + B(119895)) sdot 119860

1

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(32)

The amplitude of the additional cable force can beexpressed as

ℎ119895= 119867

2sec120579 sdot B(119895) sdot 1198601

(119895)1198602

(119895)1198603

(119895)1198604

(119895)

119879

(33)

3 Dimensionless Vibrational Mode Function

The undetermined coefficients in the solutions of the vibra-tional mode function solution for two cable segments givenby formula (32) can be determined through the boundaryconditions

1l 2l

Maei120596t Mcei120596t Mce

i120596tVce

i120596t

Vaei120596t Vbei120596t

Vcei120596t

Vcei120596tVcei120596t

Mbei120596t

120572aei120596t 120572cei120596t 120572cei120596t120572cei120596t

120572bei120596t120579aei120596t

120579bei120596t

120579cei120596t 120579cei120596t

TTTT

Figure 3 A mechanical model of a discretized cable dampersystem

As shown in Figure 3 the cable system is divided intotwo independent segments and a transverse force elementThe dynamic nodal force and dynamic nodal displacementsact on the boundaries of these models After discretizationthe nodal forces for each element include the amplitudeof the dynamic nodal shear forces 119881

119886 119881119888 and 119881

119887and the

amplitude of dynamic nodal bending moments 119872119886 119872119888

and119872119887 The nodal displacements include the amplitudes of

dynamic displacement 120572119886 120572119888 and 120572

119887and the dynamic swing

amplitudes 120579119886 120579119888 and 120579

119887 The time-varying part of these

quantities is represented by a complex exponent ei120596119905 wherei = radicminus1 is the complex unit

The boundary conditions of the displacement of the twocable segments are given by the following formula

120572119886=

1198981198921198974

0

119864119868

120593 (1205851|=0)

1205791198861198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=0)

120572119888=

1198981198921198974

0

119864119868

120593 (1205851|=1205831)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205851|=1205831)

(34)

120572119888=

1198981198921198974

0

119864119868

120593 (1205852|=0)

1205791198881198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=0)

120572119887=

1198981198921198974

0

119864119868

120593 (1205852|=1205832)

1205791198871198970=

1198981198921198974

0

119864119868

1205931015840(1205852|=1205832)

(35)

For cable segments 119895 = 1 2 each of the nodal displace-ments 120572

119886 1205791198861198970 120572119888 1205791198881198970119879 and 120572

119888 1205791198881198970 120572119887 1205791198871198970119879 is used as

boundary condition respectively to obtain the expressions

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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International Journal of

6 Shock and Vibration

for the undetermined coefficients 119860(119895)1sim 119860(119895)

4 Equation (32)

is substituted into (34) and (35) respectively Consider

120593 (120585119895) = 120589 (Φ (120585

119895) + B(119895)) (C(119895) + I sdot B(119895))

minus1

sdot 120572119886

12057911988611989701205721198881205791198881198970119879

sdot sdot sdot 119895 = 1

12057211988812057911988811989701205721198871205791198871198970119879

sdot sdot sdot 119895 = 2

(36)

where 120589 = 11986411986811989811989211989740 I = [1 0 1 0]

119879 and C(119895) are given bythe following equation

C(119895) = [Φ (120585119895|=0) Φ1015840(120585119895|=0) Φ (120585

119895|=120583119895) Φ1015840(120585119895|=120583119895)]

119879

=

[[[[

[

1 eminus119901120583119895 1 0

minus119901 119901eminus119901120583119895 0 119902

eminus119901120583119895 1 cos (119902120583119895) sin (119902120583

119895)

minus119901eminus119901120583119895 119901 minus119902 sin (119902120583119895) 119902 cos (119902120583

119895)

]]]]

]

(37)

4 Dynamic Stiffness Matrix ofthe Cable Damper System

All internal forces on cable segment 119895 can be expressed asfollows

119881(119909119895 119905) = (119864119868

1198893120593

1198891199091198953minus 119867

119889120593

119889119909119895

) sdot ei120596119905

= 1198981198921198970(120593101584010158401015840(120585119895) minus 12057421205931015840(120585119895)) sdot ei120596119905

119872(119909119895 119905) = 119864119868

1198892120593

1198891199091198952ei120596119905 = 119898119892119897

0

212059310158401015840(120585119895) ei120596119905

(38)

The equilibrium conditions for the two cable segments aregiven by the following set of equations

119881 (1199091|=0 119905) = 119881

119886ei120596119905 minus119872 (119909

1|=0 119905) = 119872

119886ei120596119905

minus119881 (1199091|=1198971 119905) = 119881

119888ei120596119905 119872 (119909

1|=1198971 119905) = 119872

119888ei120596119905

119881 (1199092|=0 119905) = 119881

119888ei120596119905 minus119872 (119909

2|=0 119905) = 119872

119888ei120596119905

minus119881 (1199092|=1198972 119905) = 119881

119887ei120596119905 119872 (119909

2|=1198972 119905) = 119872

119887ei120596119905

(39)

Equation (39) is transformed tomatrix form according tothe expressions for cable segments Consider

119881119886

119872119886

1198970

119881119888

119872119888

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205851|=0) minus 12057421205931015840(1205851|=0)

minus12059310158401015840(1205851|=0)

minus120593101584010158401015840(1205851|=1205831) + 12057421205931015840(1205851|=1205831)

12059310158401015840(1205851|=1205831)

]]]]]]]]]

]

=

119864119868

1198970

3D(1) sdot (C(1) + I sdot B(1))

minus1

sdot 12057211988612057911988611989701205721198881205791198881198970 119879

119881119888

119872119888

1198970

119881119887

119872119887

1198970

= 1198981198921198970

[[[[[[[[[

[

120593101584010158401015840(1205852|=0) minus 12057421205931015840(1205852|=0)

minus12059310158401015840(1205852|=0)

minus120593101584010158401015840(1205852|=1205832) + 12057421205931015840(1205852|=1205832)

12059310158401015840(1205852|=1205832)

]]]]]]]]]

]

=

119864119868

1198970

3D(2) sdot (C(2) + I sdot B(2))

minus1

sdot 12057211988812057911988811989701205721198871205791198871198970119879

(40)

where the matrixD(119895) is defined as follows

D(119895) =

[[[[[[[[

[

(Φ101584010158401015840(120585119895|=0)) minus 120574

2(Φ1015840(120585119895|=0))

minus (Φ10158401015840(120585119895|=0))

minus (Φ101584010158401015840(120585119895|=120583119895)) + 120574

2(Φ1015840(120585119895|=120583119895))

(Φ10158401015840(120585119895|=120583119895))

]]]]]]]]

]

(41)

The dynamic stiffness matrix of the cable segment isdefined as follows

K(119895) = 1198981198921198970120589 sdotD(119895) sdot (C(119895) + I sdot B(119895))

minus1

=

119864119868

1198970

3D(119895) sdot (C(119895) + I sdot B(119895))

minus1

(42)

Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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Shock and Vibration 7

For each cable segment the relationship between thedynamic nodal force and the dynamic displacement is writtenin matrix form as follows

K(1)

120572119886

1205791198861198970

120572119888

1205791198881198970

=

119881119886

119872119886

1198970

119881119888

119872119888

1198970

K(2)

120572119888

1205791198881198970

120572119887

1205791198871198970

=

119881119888

119872119888

1198970

119881119887

119872119887

1198970

(43)

After combining the two equations the equation forthe relationship between the total nodal force and the totaldisplacement is established Furthermore the dynamic equi-librium equation for a cable acted upon by transverse forceswith clamped boundary conditions at both ends is obtained

K(0) 1205721198881205791198881198970

=

119881119888

119872119888

1198970

(44)

where the total dynamic stiffness matrix K(0) of the cablesystem with two degrees of freedom can be obtained bycombiningK(1) andK(2)The assembly process can be writtenin matrix operator form as follows

K(0) = I1sdot K(1) sdot I

1

119879+ I2sdot K(2) sdot I

2

119879 (45)

The matrix operator in the equation is as follows

I1= [

0 0 1 0

0 0 0 1] I

2= [

1 0 0 0

0 1 0 0]K(0) (46)

K(0) can be written in the form of matrix elements asfollows

K(0) = 119864119868

1198970

3

[

[

119896(0)

11119896(0)

12

119896(0)

21119896(0)

22

]

]

(47)

where 119896(0)11= 119896(1)

33+ 119896(2)

11 119896(0)12= 119896(1)

34+ 119896(2)

12 119896(0)21= 119896(1)

43+ 119896(2)

21

119896(0)

22= 119896(1)

44+119896(2)

22 119896(1)33 119896(1)44 119896(1)43 and 119896(1)

34and 119896(2)11 119896(2)22 119896(2)

21 and

119896(2)

12corresponds to K(119895) respectivelyIt can be seen from (44) and its derivation that the equi-

librium equation takes into account the impact of bendingstiffness clamped boundary conditions and cable saggingThe factors considered here comprise a model that is most

closely related to the cables used in actual projectsThereforein theory this equation is able to describemore accurately thecable damper system in actual projects

Similarly by letting the above matrix B(119895) be zero thedynamic stiffness of each cable segment not consideringsagging is obtained as follows

K(119895) = 119864119868

1198970

3D(119895) sdot (C(119895))

minus1

(48)

The overall dynamic stiffness matrix of the cable can beobtained using (45)Thedynamic equilibrium equation of thecable can be obtained by using (44)

The horizontal dynamic stiffness of the cable with a singledegree of freedom can be obtained by rearranging (47) to thefollowing form

119896dyn = 119896(0)

11minus

119896(0)

12sdot 119896(0)

21

119896(0)

22

(49)

5 Verification

To verify the accuracy of the two analytical methods forlateral dynamic stiffness a finite element method is usedto model the cable and then a model reduction methodis used to obtain the condensed stiffness and condensedmass corresponding to the degree of freedom At this pointthe corresponding transverse dynamic stiffness of cables isobtained

Corresponding to the analytical method presented in thispaper the FEM tool-box calfem-34 which is running atMatlab environment is used to establish the finite elementmodel The cable and the damper are modeled by thetwo-dimensional plane beam element and a linear viscousdamper model respectively The FEM model takes intoaccount the effects of bending stiffness inclination angleclamped boundary conditions and sagging as well as thecontributions of the cable forces and bending moments ofthe geometric stiffness matrix of cable Reduction of massand stiffness is conducted for the two corresponding degreesof freedom on the element acted upon by transverse forcesusing Guyanrsquos static condensation Linear interpolation isthen used to obtain the transverse dynamic stiffness matrixfor the twodegrees of freedomof the subordinate target point

The basic parameters of the cable with both ends clampedwhich is used in the study are shown in Table 1

51 Transverse Dynamic Stiffness Matrix of Two Degrees ofFreedom According to the deduction process in the lastsection the corresponding transverse static stiffness of thecable can be obtained bymerely letting the circular frequencybe 120596 = 0 At this point the dynamic stiffness at the estimatedvalue of the modal frequency of the cable (provided by theformula of taut string theory in (50)) is investigated whichhas a vital significance for the actual project

120596 =

119896120587

1198970

radic119867

119898

(50)

8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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8 Shock and Vibration

Table 1 Basic parameters of the cable

Modulus of elasticity Moment of inertia Mass per unit length Area2 times 1011 Pa 8004 times 10minus6 m4 9685 kgm 124 times 10minus2 m2

Chord-wise force Length Relative location of transverse force Inclined angle3 times 106 N 200m 05 30∘

0 02 04 06 08 10

05

1

15

2

FEMSagNo sag

times107

K11

(Nm

)

1205831

(a)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K12

(Nm

)1205831

minus1

minus2

minus3

(b)

0 02 04 06 08 1

0

1

2

3

FEMSagNo sag

times104

K21

(Nm

)

1205831

minus1

minus2

minus3

(c)

0 02 04 06 08 180

100

120

140

160

180

200

FEMSagNo sag

K22

(Nm

)

1205831

(d)

Figure 4 Comparison of the terms of the static stiffness matrix along the length of cable with two transverse degrees of freedom using thethree presented methods (120596 = 0 unit Nm)

Figure 4 shows the variations in static stiffness along thecable length using the three methods The transverse axis 120583

1

represents the relative chord-wise length from the positionof the transverse force to the anchor point of the inclinedcable The vertical axis represents the stiffness term Due tothe treatment of nodal displacements and nodal forces asdescribed in the above section the four terms have the samedimension Table 2 shows a comparison of the values of thestiffness terms at the position when 120583

1is 005 01 025 05

075 09 and 095 respectively Comparison of the four termsat the midpoint of the span is carried out

As can be seen in Figure 4 the distributions of variousterms of the two-dimensional dynamic stiffness matrix havesimilar patterns in that the stiffness coefficient of a cablealong the main diagonal of the matrix is extremely large atboth ends of the cable Moving the application point of thetransverse force toward the midpoint of the cable causes thiscoefficient to decline rapidly However when 120583

1varies within

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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Shock and Vibration

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International Journal of

Shock and Vibration 9

Table 2 Comparison of values of dynamic stiffness at different locations

Method1 005 01 025 05 075 09 095

One (11987011)2

14807 times 106

72566 times 105

33728 times 105

24966 times 105

33100 times 105

71567 times 105

14695 times 106

Two (11987011) 14690 times 10

671449 times 10

532801 times 10

524356 times 10

532801 times 10

571449 times 10

514690 times 10

6

Three (11987011) 15323 times 10

674813 times 10

534502 times 10

525732 times 10

534728 times 10

575560 times 10

515478 times 10

6

Relative error3 () 33659 30044 22433 29752 46871 52844 50586One (119870

22)2 11447 11196 11067 11040 11069 11199 11450

Two (11987022) 11447 11195 11066 11038 11066 11195 11447

Three (11987022) 12729 12455 12318 12293 12328 12470 12739

Relative error () 1007 1011 1015 1019 1021 1020 1011Note 1method one is the analytic method considering sagging effects which corresponds to (42) Method two is the analytic method without consideringsagging effects which corresponds to (48) Method three is the finite element method 211987011 and 11987022 are the first and second terms on the main diagonal ofmatrix respectively 3The relative error indicates the difference between the results obtained via method one and method three

0 05 1

0

5

10

SagNo sag

1205831

minus5

120575K11

()

(a)

0 05 17

8

9

10

11

SagNo sag

1205831

120575K22

()

(b)

Figure 5 Relative error between main diagonal terms and the finite element method

[01 09] it does so smoothly in the shape of a pan Thesecondary diagonal terms transform rapidly from negativeinfinity to the smooth range in the middle of cable and thento positive infinity along the cable length

The results obtained via the three presented methodsfor calculating the transverse static stiffness of the cable areconsistent The relationship between the values obtained ineach method can be evaluated according to the followingdefinition of error

120575119896119895119895

Sag= 100 times

119896119895119895

0FEMminus 119896119895119895

0Sag

119896119895119895

0FEM

120575119896119895119895

noSag= 100 times

119896119895119895

0FEMminus 119896119895119895

0noSag

119896119895119895

0FEM

(51)

where the subscript is 119895119894 = 11 22 The superscripts 0 FEM0 Sag and 0 noSag represent the total stiffness terms

obtained using the finite elementmethod by considering andneglecting sagging effects respectively As can be seen fromTable 2 and Figure 5 the numerical differences between thestiffness terms obtained via the three presented methods arevery small Along the cable length the error between theresults of the first term on the main diagonal 119896(0)

11and the

finite element method is less than 52 119896(0)22

is also smallwithin 11Though the relative error is given byGuyanrsquos staticcondensation using finite element analysis the rigidity of thedynamic condensation itself also has errors However sincethe error of static stiffness is very small when 120596 = 0 thiscase can be used as a reference of accuracy in the study It isnoted that in this study the obtained variation pattern of thetransverse static stiffness of the cables along the cable length isreliable and has good computational accuracy The dynamicstiffness method taking sagging effects into account is evencloser to the static condensation results using finite elementanalysis

10 Shock and Vibration

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

0 05 10

05

1

FEMSagNo sag

1205831

12057311

(Nm

)

(a)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057312

(Nm

)

(b)

0 05 1

0

5

FEMSagNo sag

1205831

minus5

minus10

12057321

(Nm

)

(c)

0 05 109

095

1

FEMSagNo sag

1205831

12057322

(Nm

)

(d)

Figure 6 Comparison between the coefficient of the first-order dynamic stiffness obtained using the three presented methods (1205961199041= 2765)

Before analyzing the dynamic stiffness obtained using thethree presented methods the coefficient of dynamic stiffness120573119894119895is defined as follows

120573119894119895=

119870119894119895

0(120596119904

1)

119870119894119895

0(0)

(52)

where 1205961199041is the estimated value of the first-order modal

frequency of the cable according to the taut string equation119870119894119895

0(sdot) indicates the dynamic stiffness for a certain frequency

and119870119894119895

0(0) represents the static stiffness

Figure 6 shows the variations of the dynamic stiffness ofthe cable along its length based on the estimated value ofthe first-order modal frequency using the three presentedmethods The coefficient of the dynamic stiffness of 120573

11

corresponding to 119896(0)11

shows a distribution along the cablelength with a shape of a positively laid pan where themidpoint of the cable is a minimum having a value of

approximately 10 of the static stiffness Conversely 12057322

shows a distribution with a shape of an inverted pan wherethe midpoint of cable has a maximum value getting closer tothe static stiffness as it moves from the ends to the midpointThe nearer one is to the center point the closer the value isto the static stiffness 120573

12and 120573

21 which correspond to the

secondary diagonal show discontinuities at the midpoint ofcable Their values approach positive and negative infinityrespectively

The results of a comparison of the three presentedmethods can be drawn from Figures 6(a) and 6(b) Thedynamic stiffness obtained using the method that considerssagging and obtained by the condensation method usingfinite element analysis yields results that are close to eachother whereas the method that does not take sagging effectsinto account produces results that are quite different Thisshows that sagging effects have a significant impact on thedynamic stiffness of the cable near the first-order vibrationalfrequency and thus should be considered in calculations

Shock and Vibration 11

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

0 5 10 15

0

05

1

SagNo sagFEM

K11

(Nm

)

minus05

minus1

120596 (rad)

times107

(a)

0 5 10 15

0

500

1000

SagNo sagFEM

K22

(Nm

)

minus500

minus1000

120596 (rad)

(b)

Figure 7 Variation curve for the frequency domain of the dynamic stiffness at the midpoint of a cable (1205831= 05)

In some excitation tests of cables the excitation points ofeven-order modes are often chosen to be at the midpointsof the cables Therefore it is crucial to consider the dynamicstiffness of the cables at themidpoint in engineering vibrationtests Figures 7(a) and 7(b) show the variation curve ofthe main diagonal terms of the dynamic stiffness matrixat the midpoint of a cable within the frequency domain(ie within [0 5120587]) As can be seen from the figure thedynamic stiffness at the midpoint of the cable has an infinitediscontinuity for even order There is a zero crossing pointin each continuous interval namely the odd-order modalfrequencyThe variation curves of the dynamic stiffness withfrequencies obtained using the three presented methods arebasically consistent This shows that the calculated dynamicstiffness using the threemethods is basically consistentwithinthe investigated frequency range with very small numericaldifferences Figures 6 and 7 show that the calculated resultsgiven in this work for the dynamic stiffness at any frequencyusing the analytical algorithm for dynamic stiffness whileconsidering sagging effects are accurate and reliable

52 Transverse Dynamic Stiffness for a Single Degree of Free-dom The same conclusion can also be drawn by comparingthe transverse dynamic stiffness of the cable for a singledegree of freedom using different methods Figure 8 showsthe variations of the static stiffness along the cable length for asingle degree of freedom using three methods Figures 9(a)ndash9(d) show the variations of the coefficients of the dynamicstiffness along the cable length based on the estimatedvalue of the first- second- third- and fourth-order modalfrequencies respectively (with a reference static stiffnessvalue obtained via a finite element method) As can be seenfrom Figures 8 and 9 the analytical solution of the dynamic

times106

times104

1205831

35

3

25

2

15

1

05

00 02 04

04

06 08 1

7

68

66

64

62

05 06

SagNo sag

FEM

kdy

n(N

m)

Figure 8 A comparison between the static stiffness of cables with asingle transverse degree of freedomusing differentmethods (120596 = 0)

stiffness is accurate when considering sagging effects Theanalytical solution when sagging effects are neglected is alsoaccurate for calculating static stiffness but a large error occurswhen using this solution to calculate the first-order dynamicstiffness The static stiffness obtained by Guyanrsquos static con-densation using a finite element method gives results thatare very close to those of the former two methods Theaccuracy of the Guyan approach is acceptable for calculatingthe dynamic stiffness at fundamental frequencies Howeveras the order increases the results of the Guyan approach

12 Shock and Vibration

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

1205831

0 02 04 06 08 1

120573k

dyn

(Nm

)

0

05

1

FEMSagNo sag

(a) Mode 1

1205831

3

2

1

0

0 02 04 06 08 1

120573k

dyn

(Nm

)

FEMSagNo sag

(b) Mode 2

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(c) Mode 3

1205831

0 02 04 06 08 1

3

2

1

0

120573k

dyn

(Nm

)

FEMSagNo sag

(d) Mode 4

Figure 9 A comparison between the dynamic stiffness of cables with a single transverse degree of freedom using different methods (120596 =

120596119904

1 120596119904

2 120596119904

3 120596119904

4)

x

Displacement sensor

Vibration exciter

l0

l1

120579

y

Figure 10 Excitation test of a full-scale cable

become quite different from those obtained from the twomethods mentioned previously

This shows that static condensation method using finiteelement analysis can be appropriately used as a standard

to judge the accuracy of the calculated static stiffness butnot as the standard for dynamic stiffness In the existingdynamic condensation method using finite element analysisthe stiffness matrix of the model and the mass matrixare reduced to a specified degree of freedom at a certainfrequency point (ie the modal frequency) The question ofwhether it is suitable to be considered as a reference for thedynamic stiffness matrix at any frequency still needs to beexploredAnew topic is thus proposed namely how to realizethe reduction of dynamic stiffness at any frequency a resultwhich would play an important role in the research of theuniversal dynamic stiffness of structures

53 Vibration Tests of Full-Scale Cables To further verifythe accuracy of the analytical algorithm for the transversedynamic stiffness matrix provided in this paper on-siteexcitation data for a full-scale cable in a cable-stayed bridgeare used

The stay-cable used in the test is 2673m in length and984 cm in diameter with an elastic modulus of 2e11 Pa anda mass per unit length of 764 kgm A horizontal tension isapplied and the cable is fixed in the trough with both endsclamped and secured A TST-1000 electromagnetic vibrationexciter is used at the midpoint of the cable with powerrange of 0ndash1000N and a maximum stroke of plusmn125mm In

Shock and Vibration 13

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

0

0 50 100 150 200 250

500

0

minus500

times10minus3

Forc

e (N

)

Time (s)

Displacement

Disp

lace

men

t (m

)

Force

(a)

1

08

06120 140 160 180 200 220

400

300

200

times10minus3

Forc

e (N

)

Time (s)

Peaks of displacement

Disp

lace

men

t (m

)

Peaks of force

(b)

Figure 11 Time history of excitation test and selection of amplitude range

Table 3 A comparison between the calculated and measured dynamic stiffness

Frequency18632 20961 23290 25619 27948

Measured 1184 times 105

7609 times 104

2275 times 104

4106 times 104

1197 times 105

Method one 1157 times 105

7345 times 104

2226 times 104

3983 times 104

1159 times 105

Relative error () minus2253 minus34755 minus2153 minus30075 minus3195

Method two 1077 times 105

7153 times 104

2109 times 104

3958 times 104

1055 times 105

Relative error () minus9037 minus5992 minus797 minus3604 minus11863

Method three 1253 times 105

8062 times 105

2153 times 104

4299 times 104

1268 times 105

Relative error () 5828 5953 5363 4700 5931

the experiment a KD9200 resistance displacement meter isused to measure the displacement at the excitation point ofthe cable Figure 10 shows the layout and plan of the on-siteexcitation test

To begin the first-order natural frequency is estimatedto be 120596119904

1= 03708 lowast 2120587 using the formula for a taut string

The frequency obtained is multiplied by 08 09 1 11 and 12respectively to be used as five different work conditions inexcitation Figure 11(a) is the time history of the displacementof the excitation point at themidpoint of cable where the datais acquired by the on-site acquisition device and the timehistory of the excitation force is recorded by the excitationdevice Figure 11(b) is the curve composed of the maximumamplitude points of stable segments selected from the timehistories of displacement and force The average values aretaken to estimate dynamic stiffness

Equations (42) and (48) are used to calculate the dynamicstiffness matrix of the cable considering (method 1) andneglecting (method 2) sagging effects Equation (45) is usedto calculate the total stiffness of each matrix Then (49) isused to obtain the transverse dynamic stiffness with a singledegree of freedom which corresponds to the excitation pointof the test The calculation results are given in Table 3 Thedata in the table show due to the consideration of saggingeffects that method 1 has better precision and yields a resultthat is closer to the observed value The maximum error isless than 4Themethod that neglects sagging effects and the

finite elementmethod based on static condensation both havelarger errors This result proves that the analytical algorithmof the transverse dynamic stiffness of a cable given in thispaper has better precision and reliability

6 Conclusion

The transverse vibration of cables is a primary research focusin the field of cable dynamicsThe transverse force element isthemost common element in cable structuresThe transversedynamic stiffness of cables is a bridge between transversevibration and transverse force (excitation) Thus it is of greatsignificance for the vibration analysis of cable structures andthe vibration control of an entire project

This paper has proposed a dynamic stiffness matrix withtwo degrees of freedom at any point on a cable (transversedisplacement and cross section rotation) On this basis thedynamic stiffness corresponding to the transverse displace-ment is derived Though several factors are simultaneouslytaken into account including sagging effects inclinationflexural stiffness and boundary conditions the matrix hasa simple form and can be easily realized in programmingapplicationsThe existing studies have provided algorithms ofdynamic stiffness at the ends of cables which is incomparableto the results in this study Compared with the finite elementmethod based on static condensation our method not only

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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

14 Shock and Vibration

achieves a reasonable and accurate result for the staticstiffness but also precisely calculates the dynamic stiffnessThe variation pattern of the obtained dynamic stiffness isconsistent with the findings of the existing studiesThereforethe proposed method can be used as an effective tool instudying the transverse vibration of cables

Conflict of Interests

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

Acknowledgments

This paper was sponsored by the Project of National KeyTechnology RampD Program in the 12th Five Year Plan ofChina (Grant no 2012BAJ11B01) the National Nature ScienceFoundation ofChina (Grant no 50978196) theKey State Lab-oratories Freedom Research Project (Grant no SLDRCE09-D-01) the Fundamental Research Funds for the CentralUniversities and the State Meteorological AdministrationSpecial Funds of Meteorological Industry Research (Grantno 201306102)

References

[1] E D S Caetano Cable Vibrations in Cable-Stayed Bridges vol9 IABSE Zurich Switzerland 2007

[2] N J Gimsing and C T Georgakis Cable Supported BridgesConcept and Design John Wiley amp Sons New York NY USA2011

[3] E Rasmussen ldquoDampers hold swayrdquo Civil EngineeringmdashASCEvol 67 no 3 pp 40ndash43 1997

[4] P Warnitchai Y Fujino and T Susumpow ldquoA non-lineardynamic model for cables and its application to a cable-structure systemrdquo Journal of Sound and Vibration vol 187 no4 pp 695ndash712 1995

[5] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1975

[6] E de sa Caetano Cable Vibration in Cable-Stayed BridgesIABSE-AIPC-IVBH Zurich Switzerland 2007

[7] V Kolousek Vibrations of Systems with Curved Membersvol 23 International Association for Bridge and StructuralEngineering Zurich Switzerland 1963

[8] A G Davenport ldquoThe wind induced vibration of guyedand self-supporting cylindrical columnsrdquo Transactions of theEngineering Institute of Canada pp 119ndash141 1959

[9] A G Davenport and G N Steels ldquoDynamic behavior ofmassive guyed cablesrdquo Journal of the Structural Division vol 91no 2 pp 43ndash70 1965

[10] A S Veletsos and G R Darbre ldquoFree vibration of paraboliccablesrdquo Journal of the Structural Engineering vol 109 no 2 pp503ndash519 1983

[11] A S Veletsos and G R Darbre ldquoDynamic stiffness of paraboliccablesrdquo Earthquake Engineering amp Structural Dynamics vol 11no 3 pp 367ndash401 1983

[12] U Starossek ldquoDynamic stiffness matrix of sagging cablerdquoJournal of Engineering Mechanics vol 117 no 12 pp 2815ndash28291991

[13] U Starossek ldquoCable dynamicsmdasha reviewrdquo Structural Engineer-ing International vol 4 no 3 pp 171ndash176 1994

[14] A Sarkar and C S Manohar ldquoDynamic stiffness matrix of ageneral cable elementrdquo Archive of Applied Mechanics vol 66no 5 pp 315ndash325 1996

[15] J Kim and S P Chang ldquoDynamic stiffness matrix of an inclinedcablerdquo Engineering Structures vol 23 no 12 pp 1614ndash1621 2001

[16] W Lacarbonara ldquoThe nonlinear theory of curved beams andfexurally stiff cablesrdquo in Nonlinear Structural Mechanics pp433ndash496 Springer New York NY USA 2013

[17] A Luongo and D Zulli ldquoDynamic instability of inclined cablesunder combined wind flow and support motionrdquo NonlinearDynamics vol 67 no 1 pp 71ndash87 2012

[18] J A Main and N P Jones ldquoVibration of tensioned beamswith intermediate damper I formulation influence of damperlocationrdquo Journal of Engineering Mechanics vol 133 no 4 pp369ndash378 2007

[19] J A Main and N P Jones ldquoVibration of tensioned beams withintermediate damper II damper near a supportrdquo Journal ofEngineering Mechanics vol 133 no 4 pp 379ndash388 2007

[20] N Hoang and Y Fujino ldquoAnalytical study on bending effects ina stay cable with a damperrdquo Journal of Engineering Mechanicsvol 133 no 11 pp 1241ndash1246 2007

[21] Y Fujino andN Hoang ldquoDesign formulas for damping of a staycable with a damperrdquo Journal of Structural Engineering vol 134no 2 pp 269ndash278 2008

[22] G Ricciardi and F Saitta ldquoA continuous vibration analysismodel for cables with sag and bending stiffnessrdquo EngineeringStructures vol 30 no 5 pp 1459ndash1472 2008

[23] H M Irvine ldquoFree vibrations of inclined cablesrdquo Journal of theStructural Division vol 104 no 2 pp 343ndash347 1978

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