Experimental and Analytical Investigation of Carbon Fiber CableVibration Damping

Y.Z. Qiu & A.K. Maji

Received: 30 October 2013 /Accepted: 13 March 2014 /Published online: 4 April 2014# Society for Experimental Mechanics 2014

Abstract For structures deployed in space using cables wherevibration damping is critical for structural stability, cabledamping is significant to structural performance. To providea better understanding of damping mechanism of carbon fibercables, this paper describes the tests of cable damping underdifferent experimental configurations (different cable length,tension and type), and presents an analytical method formodeling and therefore predicting cable damping. The meth-od is developed using simplified but physically realistic as-sumptions on material constitutive properties and geometriccompatibility conditions, and considered the contact forcesand friction between helical wires. The results of the proposedmethod and several related issues are discussed and comparedwith those from experiments. These results show that theproposed method is useful and applicable for predicting cabledamping value and its variation with cable tension, length andtype of the cables.

Keywords Carbon fiber cable . Vibration . Damping .

Theoretical modeling . Laboratory test

Nomenclature

The following symbols are used in this paper:

Rc, Rw radius of the cable core and helical wiresP, P1 pitch length of helical wiresα, α1 lay angle of the undeformed and deformed

helical wires

εt total strain of helical wires along tangentdirection

εtA,εt

R the tangent strain of helical wires due toelongation and rotation

L, L1 initial and deformed length of the cablestructure

S, S1 initial and deformed length of the helical wiresθ, θ1 initial and final angle that a helical wire sweeps

outV, V1 volume of the undeformed and deformed cable

structureRh, Rh1 radius of the undeformed and deformed helical

wire centerlineΔS change of helical wire lengthΔα change of cable layer angleεc the axial strain of the cable core wire.ν Poisson’s ratioCF compaction factorθz = θ1 − θ the relative rotation of the structureδa mutual approach of wiresμ friction coefficientγz, γt ratio of the deformed and undeformed cable

structure length and helical wire lengthPnch the contact normal force between cable core and

helical wiresPnhh the contact normal force between helical wiresahh, ach contact half widthE Young’s modulusEt transverse modulusβ contact anglem number of helical wires in a layerH tangential force along helical wiresη loss factorΔU the dissipated energy per cycleU the stored energyan, an+1 two successive vibration amplitudes

Y.Z. Qiu (*) :A.K. MajiDepartment of Civil Engineering, University of NewMexico, CENT,Albuquerque, NM 87131-0001, USAe-mail: Carsonq1@unm.edu

A.K. Majie-mail: amaji@unm.edu

Experimental Mechanics (2014) 54:1087–1097DOI 10.1007/s11340-014-9878-y

δ logarithmic decrementξ cable modal damping ratioΔ1 relative movementS6T tonsorial shear strain between cable

wires associated with wire axial strainUε stored strain energyy vibration profileb vibration amplitudeUT stored energy due to tension

Introduction

The vibrating taut string or cable was one of the first physicalsystems to which the analytical tools of mechanical and math-ematics were applied. The problem of cable dynamics and thecharacterization of cable damping still attract a great deal ofattention from the scientific community leading to a richtechnical literature. This is due to the wide application ofcables in engineering fields and to their tendency to vibrate.For aerospace large lightweight structures in space, undampedvibrations can be problematic and may lead to malfunctions.Consequently, how much energy can be dissipated by cablesis of importance when considering dynamic behavior of struc-tures with cables.

Many researchers have tested cables in attempts to under-stand the behavior and the dependency of cable damping oncable tension, length, and construction (type). Hard andHolben [1] investigated the self-damping of tensioned trans-mission line conductors and found that the cable length has noeffect on cable damping. However, damping decreases as thetension increases. Wei and Kukureka [2, 3] employed theresonance technique to evaluate the damping of optical fibercables. Their data indicates that cable damping first decreasesas length increases, and then reaches a stable value. Barbieriet al. [4] designed a non-contact vibration test set-up fortransmission line cables. Their results demonstrate that cabledamping decreases as length and tension increases. Rambergand Griffin [5] measured the damping of a taut pinned cableand obtained results similar to that of Barbieri et al. [4].

In many of the earlier theoretical analyses, several simpli-fying assumptions have been made to obtain analytical orclosed form solutions. One of the assumptions is about theinterwire friction, which takes place at the interfaces of cablewires and is hard to model. For simplification, most of theresearchers analyzed two extreme cases: either a no-slip fric-tion model or a full-slip friction model. Machida and Durelli[6], Chi [7, 8], Knapp [9], Kumar and Cochran [10] investi-gated cable damping but neglected the effect of interwirefriction. An attempt to relate the internal friction to thedamping properties of a cable was made by Vinogradov andAtatekin [11]. Claren and Diana [12, 13] analyzed the internaldamping of axially loaded stranded cables by introducing the

slippage coefficient. However, no further work to formulate thecable internal damping and slippage coefficient was provided.Hobbs and Roof [14] addressed the problem of energy dissi-pation in multilayered spiral strands, but simplified their modelbased on homogenization of the cable layers into orthotropiccylindrical sheets. Moreover, the integration of contact forcesinto the cable damping mechanics remains a challenge. Tosimplify the problem, usually only one type of cable contact,cable core-wire contact or wire-wire contact, is considered, andinterwire friction is neglected or assumed to be a constant.Sathikh [15], Labrosse and Conway [16, 17], Leech [18, 19]studied the interwire friction effects on seven-wire cablesconsidering only the wire-core contact of the cable.

All of these investigations enriched the understanding ofthe cable damping behavior. However, the internal dampingmechanism of cables is still not fully understood. Some ofthese results are also apparently contradictory. For example,Hard and Holben [1], Wei and Kukureka [2, 3], Ramberg andGriffin [5] provided different conclusions for the dependencyof cable damping on cable length. Moreover, the commonlyused ‘thin rod’ model [15–17] modeled the cable wire as a‘thin rod’ or ‘curved beam’ within the framework of beamtheory which is not appropriate for tensioned cable. The‘orthotropic sheets’ theory proposed by Raoof [14] modeledthe cable wire layers as several cylindrical orthotropic sheets,however, it is based on the homogenization of the cable layersinto orthotropic cylindrical sheets and is not appropriate forcables with one core and six helical wires. In addition, most ofthe investigations focus on the damping behavior of steelcables, while the available investigations on carbon fiber cabledamping is scarce. Thus, the objective of this study was toconduct a series of carefully instrumented tests on carbon fibercables to get a better understanding of the damping properties,and then establish a model by including the cable interwirecontact and interwire friction to predict the damping propertiesof carbon fiber cables.

Experiments and Results

The vibration damping of cables in different lengths (0.2032,0.304 and 0.5080 m), tensions (111.25, 222.50, 333.75,445.00 and 578.50 N) and constructions (type) (20.71, 41.42and 62.13 turns/m) were tested using the experimental set-up,as shown in Fig. 1, and procedures described by Maji and Qiu[20, 21]. The experimental set-up presented in Fig. 1 consistsof a wooden frame with two steel plates that can clamp thecables to maintain the applied tension force. A load cell (PCBModel 352A73) and a data acquisition equipment (NI USB9234) are connected to a laptop for signal collecting andprocessing. The tested cables are made by twisting sevenIM7 carbon fiber (HERCULES INC, type IM7-W-12K) towsfor a certain number of turns. For instance, to fabricate a

1088 Exp Mech (2014) 54:1087–1097

0.3048 m carbon fiber cable, the untwisted strands are firstfixed at left end, and then the strands are twisted for 10, 20 and30 turns to form cables in three different configurations. Thecables are then is identified by the number of turns per meter,20.71 turns/m, 41.42 turns/m and 62.13 turns/m, respectively.For each cable, seven independent and well organized testswere conducted, and 14 data (from 2 accelerometers of seventests) recorded by the two accelerometers locate at 1/5 and 3/4of the cable length were used to determine the damping ratiousing the ‘half-power bandwidth’ method in the frequencydomain. The damping values obtained by this method forthose tests were then averaged to give the damping valuesshown in Table 1 for the measures cables.

The damping of tested cables in different lengths andconstructions undergoing varied tension force are presentedin Table 1, and are plotted as Figs. 2, 3, and 4. The results inFig. 2 demonstrated that higher tension results in lowerdamping. Figure 3 showed that for the carbon-fiber cablestwisted by different turns, as the number of twists in the cableincreases, the damping decreases. This could be because whenthe number of twist increases, the cable becomes tighter, andreducing the movement and friction internal to the cable. Forthe effect of cable length on cable damping, Fig. 4 showed thatas the cable length increases, the damping decreases, butbeyond a particular value, the damping remains unchanged.

Analytical Investigation of Cable Damping

Description of Cable Geometry

The tested carbon fiber cables are assemblies of millions ofIM7 fibers and are fabricated by twisting carbon fiber tows.For simplification, let us consider the cable as a 1+6 cable

structure, as shown in Fig. 5, in which the cable has 1 centralstraight fiber core and 6 helical wires. It should be noted thatthis geometry usually represents the metallic cables, and thetested carbon fiber cables can be approximately representedby this geometry. The geometry of each helical wire is char-acterized by the pitch length, P, which is the reciprocal of twistper unit length, and the lay angle, α, measured with respect tothe axis of the cable (Z axis). The helical wire centerline is ahelical curve of radius Rh related to Rw the radius of the helicalwires, and Rc the radius of the cable fiber core:

Rh ¼ Rc þ Rw ð1Þ

The pitch length of the cable was determined by Costello[22] as:

P ¼ 2πRh

tan αð Þ ð2Þ

Where:

Rh initial radius of helical wires centerlineRc,Rw radius of cable core and helical wiresP pitch length of helical wiresα initial lay angle of helical wires

Considerations

a) Displacements and strains are assumed to be small;b) The cable wires have a coupling behavior between exten-

sion and twisting, and the helix angle variation is considered;c) The cable core is assumed to have a circular cross-section,

and the helical wires are assumed to have elliptical cross-section due to the process of manufacture of the cables;

d) The cable is considered to be comprising of a core and sixhelical wires, and both core helical wire interaction andinteraction among helical wires are considered in theanalysis;

e) Effect of Poisson’s ratio and the contact deformation areconsidered.

Axial Strain of Helical Wires

In axial loading, with traction and torsion, the axial strain ofeach cable wire is assumed to have two parts: the first part

Fig. 1 Experimental set-up

Table 1 Cable properties

Cable No Lay angle Poisson’s ratio Young’s Modulus (Gpa) Helical centerline radius (m) Friction coefficient

1 0.1207 0.3 2.76E + 11 0.0004658 0.4

2 0.2332 0.3 2.76E + 11 0.0004562 0.4

3 0.3328 0.3 2.76E + 11 0.0004425 0.4

Exp Mech (2014) 54:1087–1097 1089

results from the elongation of the overall cable, whereas thesecond part is due to its rotation. For small deformation, thestrain of the cable wires can be expressed as:

εt ¼ εAt þ εRt ð3Þ

Where:

εt total strain of the helical wires along the tangentdirection, t designates the tangent direction of thehelical wires

εtA,εt

R the tangential strain of the helical wires due toelongation and rotation

Axial strain due to elongation

Let γz be the extension ratio, i.e. the deformed structure’slength to the initial structure lengthmeasured along the cable’saxis (z-axis), and γt be the corresponding extension ratio for ahelical wire whose initial and final radii are Rh and Rh1,respectively. As shown in Fig. 6, we have:

γz ¼L1L

ð4Þ

γt ¼S1S

¼ 1þ εAt ð5Þ

Where:

L, L1 initial and deformed length of the cable structureS, S1 initial and deformed length of the helical wires

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

theoretical,0.2032mexperimental,0.2032mtheoretical, 0.3048mexperimental, 0.3048mtheoretical,0.5080mexperimental, 0.5080m

(a)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

theoretical,0.2032mexperimental,0.2032mtheoretical, 0.3048mexperimental, 0.3048mtheoretical,0.5080mexperimental, 0.5080m

(b)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

theoretical,0.2032m

experimental,0.2032m

theoretical, 0.3048m

experimental, 0.3048m

theoretical,0.5080m

experimental, 0.5080m

(c)

Fig. 2 Comparison of cable damping (a) 20.71 turns/m (b) 41.42 turns/m, (c) 62.13turns/m cables

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

20 25 30 35 40 45 50 55 60

Dam

pin

g (

%)

Construction (turns/m)

111.25N 222.50N333.75N 444.5N578.50N

(a)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

20 25 30 35 40 45 50 55 60

Dam

pin

g (

%)

Construction (turns/m)

111.25N 222.50N333.75N 444.5N578.50N

(b)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

20 25 30 35 40 45 50 55 60

Dam

pin

g (

%)

Construction (turns/m)

111.25N 222.50N333.75N 444.5N578.50N

(c)

Fig. 3 Cable damping versus cable construction (a) 0.2032 m (b)0.3048 m (c) 0.5080 m cables

1090 Exp Mech (2014) 54:1087–1097

θ, θ1 the initial and final angle that a helical wire sweepsout in a plane perpendicular to the cable axis

As the helical wires are extended, the cable strands undergoa compaction of volume. The ratio of the deformed and un-deformed volume of the cable is defined as the compactionfactor (CF):

γz ¼L1L

¼V 1

πR2h1V

πR2h

¼ CFR2h

R2h1

ð6Þ

Where:

V, V1 the volume of the un-deformed and deformed cableCF compaction factor of the cableRh,Rh1

radius of the un-deformed and deformed helicalwires

From equation (2), we have:

tanα ¼ 2πRh

Pð7Þ

Therefore, the pitch length in the deformed state can bedetermined by:

P1 ¼ Pγz ð8Þ

From equations (7) and (8), the lay angle of the deformedstate can be determined as:

tan α1 ¼ tanα

γ3=2z

ð9Þ

The initial length of a cable helical wire with lay angle α is:

S ¼ L

cos αð10Þ

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Dam

pin

g (

%)

Length (m)

111.25N222.50N333.75N444.5N578.50N

(a)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Dam

pin

g (

%)

Length (m)

111.25N222.50N333.75N444.5N578.50N

(b)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Dam

pin

g (

%)

Length (m)

111.25N222.50N333.75N444.5N578.50N

(c)

Fig. 4 Cable damping versus length (a) 20.71 turns/m (b) 41.42turns/m(c) 62.13turns/m cables

Fig. 5 Cable geometry

Fig. 6 Developed view of helical wire centerline

Exp Mech (2014) 54:1087–1097 1091

From equation (4), the axial length L1 corresponding to thedeformed state is:

L1 ¼ Lγz ð11Þ

Hence, the corresponding helical wire length in the de-formed state is:

S1 ¼ L1cos α1

¼ L*γzcos α1

ð12Þ

Where:

α1 lay angle of the deformed helical wires

Using equations (5), (9), (10) and (12), the helical wireextension ratio can be expressed as:

γt ¼ γzcos αcos α1

¼ γ2z cos2αþ sin2α

γzð13Þ

From equations (5) and (13), the axial strain of the cablewires due to tension can be obtained as:

εAt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2z cos

2αþ sin2αγz

s−1 ð14Þ

From equation (14), the axial strain of helical wiresis a function of the initial lay angle, initial radius ofhelical wires centerline and the radius of deformed cablehelical wires centerline. The lay angle can be deter-mined from the cable construction defined in Section 2(turns/m) using equation (7). The radius of the deformedcable helical wires centerline will be discussed inSection 3.4.1 by considering the contact deformationand Poisson’s ratio.

Axial strain due to rotation

A relative rotation θz exists between the undeformedand deformed cable end section. The axial strain ofthe helical wires due to this rotation can be expressedas:

εRt ¼ ΔS

Sð15Þ

Where:

ΔS ¼ Rh*θz*sinα1 ð16Þ

Substituting equation (6), (10) and (16) into equation (15),the axial strain of the helical wires due to rotation can beexpressed as:

εRt ¼ffiffiffiffiffiffiffiCF

p Rhffiffiffiffiffiγz

p θzLsinα1cosα ð17Þ

RhθzL

¼ Rh 1þ εcð ÞRh1cotα1

−1

cotαð18Þ

Where:

θz=θ1−θ the relative rotation of the helical wires

Substituting equation (18) into (17), the axial strain due torotation can be rewritten as:

εRt ¼ffiffiffiffiffiffiffiCF

pffiffiffiffiffiγz

p Rh 1þ εcð ÞRh1cotα1

−1

cotα

� �sinα1cosα ð19Þ

Where:

εc the axial strain in the cable core

Assuming small deformation:

Δα ¼ α1−αj j≪1 ð20Þ

Hence cosα1 can be expressed as:

cosα1 ¼ cos αþΔαð Þ ¼ cosα−Δαsinα ð21Þ

The axial strain of a straight cable core εc in equation (19) istherefore:

εc ¼ L1−LL

¼ 1þ εtð Þcosα1

cosα−1 ð22Þ

Substituting equation (21) into equation (22), andneglecting the higher order terms, equation (21) can now bewritten as:

εc ¼ εt−Δαtanα ð23Þ

From equation (9), we have:

cotα1 ¼ γ3=2z

tanαð24Þ

sinα1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tan2αγ3z þ tan2α

sð25Þ

1092 Exp Mech (2014) 54:1087–1097

Substituting equations (14), (19), (23) and (25) into equa-tion (3), the total strain of the helical wires along the tangentdirection can be written as:

εt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2z cos

2αþ sin2αγz

s−1þ

ffiffiffiffiffiffiffiCF

γz

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ3z cos

2αþ sin2αp

* 1−Δαtanαð Þ�ffiffiffiffiffiffiffiCF

p γ3=2z

tanα

−1

cotα

2664

3775

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan2α

γ3z þ tan2α

scosα ð26Þ

From equation (26), the total strain of the helical wiresalong the tangent direction can be determined as a functionof the initial lay angle α, initial radius of helical wires center-line Rh, radius of the helical wires centerline of the deformedcable Rh1, the compaction factor (CF) and the change of thelay angle Δα. Determination of Rh1 and Δα are presented inthe following Section.

Modeling of Contact and Friction Forces

Radius of deformed helical wires centerline

Contact forces between cable wires result in deformation thatreduces the radius of the helical wires and consequentlyreduces the equilibrium contact force per unit length andtension resulting in the wires corresponding to specific strandstrains. If the contact deformation in the cable core and helicalwires is considered, and if the Poisson’s ratio effects in theindividual wires are considered, the final helical radius cannow be written as:

Rh1 ¼ Rc 1−νεcð Þ þ Rw 1−νεtð Þ−δa ð27Þ

Where ν = Poisson’s ratio of the material, and δa is themutual approach of the cable core and cable helical wires,which can be determined from contact theory. The mutualapproach between two parallel cylinders is given by Roarkand Young [23] as:

δa ¼ 2Pnch 1−ν2ð ÞπEt

2

3þ ln

4Rc

aþ ln

4Rw

a

� �ð28Þ

Therefore the final helical wire centerline radius Rh1 is:

Rh1 ¼ Rc 1−νεcð Þ þ Rw 1−νεtð Þ−2Pnch 1−ν2ð ÞπEt

2

3þ ln

4Rc

acþ ln

4Rw

ac

� �

ð29Þ

Where:

Et, transverse modulus

Interwire contacts and friction

Strands in cables may subject to core-wire, wire-wire orcoupled core-wire and wire-wire contacts depending on theconstruction of the strand and the type of loading. The contactmodels may change from one model to the other depending onthe force and the deformation of the core and wires. Most ofthe literature on vibration damping analyzes the cable strandwith either the core-wire contact or wire-wire contact. In thispaper a coupled core-wire and wire-wire contact is consideredto understand the effect of interfacial forces on the cablevibration damping. Furthermore, the friction at the interfacewas also included in this model. In the portion where slipoccurs the helix angle will change. This increase in helix angleunder loads is determined from equations (21) to (23) byconsidering the small deformation:

Δα ¼ 2−εtð Þtanα ð30Þ

In a simple cable, the contact zone between a helical wireand the core forms a narrow strip whose central line is a helix.This contact can be locally approximated as the contact be-tween two parallel straight cylinders. Because the contactwidth is very small compared with the wire radius, theHertzian contact theory is applicable. The contact half-widthbetween a core and a helix, ach was determined by Labrosseand Conway [16, 17] as following:

ach ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1−ν2ð ÞRwRcPnch

πE Rw þ Rcð Þ

sð31Þ

Similarly, the contact half width between two helical wiresahh is:

ahh ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1−ν2ð ÞRwRcPnhh

πE Rw þ Rcð Þ

sð32Þ

Exp Mech (2014) 54:1087–1097 1093

Where:

Pnch the normal load per unit length between cable core andhelical wires

ach the contact half-width of cable core and helical wirescontact

Pnhh the normal load per unit length between helical wiresahh the contact half-width of helical wires contactE Young’s modulus of the cable material

Contact normal load per unit length

Along the contact line between the helical wires, the normaldistributed force Pnhh and the tangential distributed forcesexist. In addition, along the line of contact between the cablecore and a helical wire, the normal distributed force Pnch andthe tangential distributed forces exit. The contact angle β,which defines the direction of the distributed contact loadPnch, is given by Costello and Phillips [24] as following:

cosβ ¼ 1

cos2α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

tan2π2−πm

� sin2α

vuut−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan2

π2−πm

� 1þ 1

tan2α*cos2π2−πm

� sin2αþ tan2

π2−πm

� h8><>:

9>=>;þ sin4α

vuuuut8>><>>:

ð33Þ

Where:

m the number of helical wires, equal to six for a 1+6 strand

As showed by Hobbs and Raoof [14], when the changes inhelix angle and radius are small, the interwire distributed forcesin the normal direction can be represented approximately as:

Pnch ¼ −Hsin2α=Rh ð34Þ

Where:

H ¼ πR2wEεt ð35Þ

Substituting equation (35) into equation (34), the distribut-ed contact force between the cable core and helical wires,denoted by Pnch, can be rewritten as:

Pnch ¼ −πR2wEεtsin

2α=Rh ð36Þ

The contact force between the helical wires can be deter-mined to be:

Pnhh ¼ −Pnch

2cosβ¼ πR2

wEεtsin2α

2Rhcosβð37Þ

Determination of Vibration Damping

An energy based method is used to evaluate the cable vibra-tion damping. For low damping, the energy stored at twosuccessive peak amplitudes, an and an+1, of a freely decayvibration is related to the loss factor by Raoof and Huang [25]:

η ¼ ΔU

U≈a2n−a2nþ1

a2n≈2 an−að nþ1Þ

anþ1ð38Þ

Where:

η loss factor of the systemΔU the energy dissipation per cycleU the stored energyan, an+1 two successive vibration amplitudes

The logarithmic decrement δ of the cable system is givenby Chopra as [26]:

δ ¼ lnananþ1

� �ð39Þ

For low damping, the following expression can be obtained:

δ ¼ η2≈an−að nþ1Þanþ1

¼ 1

2

ΔU

Uð40Þ

Then damping ratio of the cable system ξ is determined by:

ξ ¼ δffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π2 þ δ2

p ¼ΔU

Uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16π2 þ ΔU

U

� 2q ð41Þ

Considering two adjacent helical wires of an axiallypreloaded carbon fiber cable, sliding takes places while thetangential friction forces remains atμfs. The tangential relativedisplacement between central wires is estimated by Raoof[27], based on the parametric studies, and is given as:

Δ1 ¼ 4RwS6T ð42Þ

S6T ¼ εt 0:00196α−0:000394α2 þ 0:0000247α3�

;α≤25�

ð43Þ

1094 Exp Mech (2014) 54:1087–1097

Where:

Δ1 tangential relative displacementS6T shear strain between cable wires associated with wire

axial strain

Then friction energy dissipation of helical wires is deter-mined as:

ΔU ¼ mμf s εt þΔ1ð Þ ð44Þ

Where fs is the contact normal force determined by equa-tions (36), (37) and the associated contact half width, and m isthe number of helical wires.

The stored strain energy can be determined as:

Uε ¼Z l

0

1

2EAε2t ds ð45Þ

Assuming that the sinusoidal vibration profile is:

y ¼ bsinπxl

ð46Þ

For small amplitude of vibration, axial tension can beapproximated to be a constant, and the strain energy due tothe tension can be expresses as:

UT ¼ 1

2

Z l

0y0Tdx ¼ b2

2

π2

l2T

Z l

0cos

πxl

� 2dx ð47Þ

Where:

UT initial stored energy due to the applied tension force

Y the assumed sinusoidal cable vibration profileb vibration amplitude

The stored energy of the cable is then expressed as:

U ¼ UT þ Uε ð48Þ

The damping can then be determined using equations(38)–(48). This iterative procedure was implemented usingan iterative MatLab Script.

Results and Discussion

The carbon fiber cables tested in the laboratory were analyzedusing the proposed method. Table 2 shows the data pertinentto these cables. The length of cable is 0.2032, 0.3048, and0.5080 m, respectively, and the transverse modulus of thecable is assumed as 8 % of the longitudinal tensile modulus,which is the typical value for IM7 carbon fiber. The results ofcable damping obtained using this proposes analytical methodreported earlier is shown in Table 3.

Figure 2 shows how the analytical results compare with theexperimental data as a function of tension. The threeFigures (Fig. 2(a–c)) correspond to the three different cableconstruction (# of turns). In each Figure the variation ofdamping with applied tension and length of the cable ispresented. It can be seen that the analytical model is capableof capturing the trends on the variation of damping corre-sponding to each of the three independent variables (length,applied tension and construction).

Table 2 Experimental results of cable vibration damping

Tension (N) Damping (%)-20.71 turns/m Damping (%)-41.42 turns/m Damping (%)-62.13 turns/m

0.2032 m 0.3048 m 0.5080 m 0.2032 m 0.3048 m 0.5080 m 0.2032 m 0.3048 m 0.5080 m

111.25 5.50 2.42 2.22 4.59 2.27 1.54 4.59 1.68 1.14

222.50 3.79 2.11 1.81 3.07 1.80 1.11 3.07 1.24 0.96

333.75 3.18 1.83 1.33 2.53 1.68 0.94 2.53 1.18 0.54

445.00 3.06 1.45 1.14 2.35 1.39 0.68 2.35 0.97 0.50

578.50 2.60 1.40 1.00 2.18 1.36 0.70 2.18 0.93 0.36

Table 3 Theoretical analysis results of cable vibration damping

Tension (N) Damping (%)-20.71 turns/m Damping (%)-41.42 turns/m Damping (%)-62.13 turns/m

0.2032 m 0.3048 m 0.5080 m 0.2032 m 0.3048 m 0.5080 m 0.2032 m 0.3048 m 0.5080 m

111.25 6.09 4.11 2.81 5.02 3.19 1.92 4.67 3.11 1.87

222.50 3.72 2.52 1.77 2.89 1.82 1.09 2.62 1.75 1.05

333.75 2.67 1.82 1.30 2.02 1.27 0.76 1.83 1.22 0.73

445.00 2.09 1.42 1.02 1.56 0.98 0.59 1.40 0.93 0.56

578.50 1.65 1.13 0.81 1.22 0.76 0.46 1.09 0.73 0.44

Exp Mech (2014) 54:1087–1097 1095

Figure 2 shows that the analytical model captures thedecrease in damping as tension increases. Typically, the mea-sured damping agrees well with the proposed model estima-tion. Figure 2 also shows that the measured damping ratios donot have a strong dependency on cable tension force as thecalculations when the tension force is lower, which indicatedthat for lower tension, the damping mechanism of cable ismuch more complicated, and this could be attributed to thetransverse modulus of the cable is likely to increase as moretension is applied, which was not considered in the proposedmodel. Also, for cable subjected to lower tension force, theinterwire slip might not the main damping mechanism, andwhich is not the focus of this paper.

Also, the measured damping ratios were given as averagevalues from several well organized and configured indepen-dent tests. Considering that the overall damping is dominatedby the interwire friction (in the sliding phase) as shown in thecalculation, the measured damping estimation is appropriatedfor cable damping with nonlinear properties.

The analytical damping results provided in Table 3 arepresented in graphical form in Figs. 3 and 4 against with cableconstruction and length, respectively. The trends comparevery well with the test data of Section 2.

Figure 7 shows the sensitivity of the analytical value ofcable damping to the Poisson’s ratio. It can be noted that thedamping values are not very sensitive to the Poisson’s ratio forsmall and reasonable assumptions (0.3 used in this analysis,typical of carbon fibers). With a higher Poisson’s ratio, theflattening of the cable causes more contact and friction be-tween cable wires which leads to higher energy dissipation.

To investigate if it is necessary to include the change of layangle in the model, Fig. 8 presents the theoretical identifieddamping with (Δα≠0) and without (Δα=0) consideration ofthe change of lay angle, and analytical results are comparedwith the test results. It can be seen that without taking intoaccount of the change of lay angle, the analytical modelunderestimates cable damping (around 50% decrement ratio).

Conclusions

Carbon fiber cables were tested under different experimentalconfigurations using the designed experimental setup. Fromthe test results, the cable damping decreases with the incre-ment of tensile force and the number of twists. Also, as thecable length increases, the cable damping decreases until aconstant damping ratio is reached.

An analytical method has been developed which allows thedamping of an axially loaded carbon fiber cable to be predict-ed with reasonable accuracy, based on the properties of thecable. The frictional energy dissipation is considered to be themain source of cable damping. Information has also beenprovided for estimating the extensional-torsional axial strain,

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 0.1 0.2 0.3 0.4 0.5

Dam

pin

g (

%)

Poisson's ratio

111.25N 222.50N333.75N 444.50N578.50N

Fig. 7 Cable damping versus Poisson’s ratio for 20.71 turns/m, 0.2032mcarbon fiber cable

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

Δα=0,theoretical

Δα≠0,theoretical

test

(a)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

Δα=0,theoretical

Δα≠0,theoretical

test

(b)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 100 200 300 400 500 600

Dam

pin

g (

%)

Tension (N)

Δα=0,theoretical

Δα≠0,theoretical

test

(c)

Fig. 8 Comparison of damping (with and without consideration ofchange of lay angle) (a) 20.71 turns/m, 0.2032 m (b) 20.71turns/m,0.3048 m (c) 20.71turns/m, 0.5080 m cables

1096 Exp Mech (2014) 54:1087–1097

the wire flattening and the contact behavior. In particular, thecoupled wire-wire contact and wire-core contact, and thechange of lay angle were considered in the model. The ana-lytical model captured the trends on the variation of dampingcorresponding to each of the three independent variables(length, applied tension and construction) as those of tests.

In summary, this paper provides a method for experimen-tally determining damping values for carbon fiber cables forvarious applied conditions (length, tension, constructions,etc.). This information can be used in the analysis of thedynamics of cable-supported structures. In addition, somestraightforward formulations were presented which shouldprove useful in predicting cable vibration damping. However,the accuracy and appropriateness of the model is affected bythe simplifying assumptions and is also affected by themanufacturing process.

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