Journal of Sound and Vibration (1995) 181(1), 71–83

ENERGY-BASED EVALUATION OF MODALDAMPING IN STRUCTURAL CABLES WITH

AND WITHOUT DAMPING TREATMENT

H. Y R. A

Department of Civil Engineering, Saitama University, 255 Shimo-Okubo, Urawa,Saitama 338, Japan

(Received 19 Janaury 1993, and in final form 6 December 1993)

The modal damping characteristics of single structural cables are analyticallyinvestigated in this paper. An energy-based representation of modal damping in structuralcables is first derived in the form of the product of modal strain energy ratio and loss factor.The ratio of the modal strain energy to the total potential energy associated with modalvibration is next calculated numerically for both axial and bending deformation, byapplying a finite element method, and the characteristics of each strain energy ratio in eachmode of the structural cable is studied. It is deduced from the present analysis that themodal damping in a structural cable is generally very low because of the very largecontribution of the initial cable stress to the total potential energy, causing very small modalstrain energy ratios. The modal damping of the damping treated structural cable [1] is alsoestimated by using the same energy-based representation. The performance and theeffectiveness of this damping treatment on structural cables is finally discussed, based onthe estimated values of the modal damping.

1. INTRODUCTION

Over the past several decades, cables have gained wide popularity as structural membersbecause of their light weight and high strength. Recent advances in structural andconstructional technologies have also enabled engineers to use the cable efficiently inrelatively large structures such as long span cable-supported bridges. However, it is wellknown that structural cables, because of their high flexibility, light weight and low dampingcharacteristics, are easily excited into severe oscillations through the dynamic effects ofwind or rain–wind interaction. In order to predict the dynamic behavior of a singlestructural cable, or of the whole cable structure, estimation of modal damping isindispensable because the dynamic response depends significantly on the modal dampingratios. However, it is difficult to estimate analytically the modal damping of cablestructures. As a result, the present day practice of estimating the damping performanceof cable structures, such as cable-stayed bridges, is based on an empirical approach basedon field experimental data, rather than on an analytical approach.

In this paper, the energy approach is adopted to evaluate analytically the modaldamping of a single structural cable. The evaluation of the total potential energy and thestrain energy associated with each vibration mode gives one the modal damping, basedon the loss factor. The loss factor is defined, for the possible deformation of the cable,as the ratio of the energy dissipated to the elastic energy stored per cycle of vibration. Theratio of the modal strain energy to the total modal potential energy, which appearsexplicitly in the energy-based representation of modal damping of the structural cable, is

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0022–460X/95/110071+13 $08.00/0 7 1995 Academic Press Limited

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Figure 1. A cross-section of a damping-treated structural cable.

calculated numerically for both axial and bending deformations, by a finite elementmethod. The characteristics of each strain energy ratio, in each mode of the structuralcable, is first studied based on the analytical results. The modal damping of the structuralcable is then discussed.

This energy-based approach is next extended to estimate the modal damping ofstructural cable with a damping treatment, previously studied in reference [1]. In thedamping treatment proposed in the reference, a layer of viscoelastic material is supposedto be introduced between the existing outer cover and the inner wire strands of a structuralcable, as shown in Figure 1. The main idea of the damping treatment is to increase themodal damping of structural cables at the material level, in order to reduce cable vibrationsto serviceable limits, without the use of any external damping devices [2], or without theapplication of any extra external energy supply, as required in active control [3]. Thematerial damping imparted by such a damping treatment applied to structural cables wasinvestigated in reference [1], in terms of the loss factor. In the present paper, theenergy-based approach is applied to evaluate the structural damping of a single suspendedstructural cable in order to investigate the effectiveness of such a damping treatment interms of modal damping. Only the increment in the modal damping ratio induced by thedamping treatment is estimated, it being assumed, as in reference [1], that the introductionof such viscoelastic material does not alter the original damping characteristics of thestructural cable. The effectiveness of the damping treatment is then discussed, based onthe estimated value of the modal damping increment.

2. EVALUATION OF MODAL DAMPING RATIOS

Modal damping is defined as the damping associated with each mode of vibration ofa damped system. According to the energy-based definition of the damping ratio [4], themodal damping for the nth mode (jn ) is evaluated as the ratio of the modal dissipatedenergy per cycle (Dn ) to the modal potential energy (Un ):

jn =Dn /4pUn . (1)

Cables in cable structures are applied with initial tension in order to maintain the staticequilibrium. This initial tension gives rise to the geometrical stiffness which constitutes adominant part in the total stiffness of the cable. In the case of structural cables undervibration, the initial tension in the cable also performs work against the dynamic strainin the cable, which is equivalent to the potential energy associated with the geometricalstiffness [5]. Therefore, in the case of cables, the total potential energy (Un ) is composedof the modal potential energy of the initial tension in the cable (Utn ), the modal strainenergy due to the axial deformation (Uan ) and the modal strain energy due to the bendingdeformation (Ubn ): that is,

Un =Utn +Uan +Ubn . (2)

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Note that the bending deformation is also taken into account in this paper to evaluatethe modal damping of the cable, while the bending stiffness is generally neglected in themechanics of ordinary structural cables. This is because the bending deformation mightcontribute to some extent to the total modal damping of the cable, especially when thecable has a damping treatment.

As for the modal dissipated energy in the structural cable (Dn ), there can be varioussources of energy dissipation. However, since the main objective of the present study isto investigate the inherent damping characteristics of structural cables, only the internalsources are considered, and the external sources of dissipation, such as aerodynamicdamping, are neglected. Two internal sources of energy dissipation can be identified in thecase of flexural vibration of cables: the energy dissipated by the axial deformation (Dan )and the energy dissipated by the bending deformation (Dbn ). Although the energydissipation by the axial deformation (Dan ) is the major source of internal damping in thecase of ordinary structural cables [6, 7], the energy dissipation due to the bendingdeformation (Dbn ) may not be negligible in the case of damping treated cables because ofthe larger value of the bending loss factor [1]. The loss factor h is one of the dampingparameters and is defined by the ratio of the dynamically dissipated energy to the strainenergy stored per cycle [8]. Upon introducing this loss factor for each deformation, themodal energy dissipation associated with the axial and the bending deformations arerepresented by

Dn =Dan +Dbn =2p(Uanha +Ubnhb ), (3)

where ha and hb are the axial and bending loss factors of the structural cables.Substituting equations (2) and (3) into equation (1) gives the final form of the expression

for the nth modal damping as

jn = 12 0Uan

Unha +

Ubn

Unhb1 . (4)

From equation (4), it is evident that the evaluation of the modal potential energyratios, as well as the loss factors, is indispensable for the estimation of modal dampingand for the investigation of the effectiveness of the damping treatment in structuralcables.

3. MODAL POTENTIAL ENERGY IN STRUCTURAL CABLES

3.1.

Structural cables are initially stressed in order to support other structural members. Thestatic equilibrium configuration of such an initially stressed structural cable is chosen hereas a reference state of the cable dynamics. The variational principle for initially stressedproblems, given in reference [5], is then directly applied to the present problem. Upondenoting the initial stress (induced in the cable by the distributed load f0i per unit arc lengthof the cable) by s0 and assuming no bending stiffness in the cable, the static equilibriumequation for the initial configuration xi is then given as [9]

dds 0As0

dxi

ds1+ f0i =0, i=1, 2, 3, (5)

where s is the initial arc length and A is the cross-sectional area of the cable.

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Now consider an additional dynamic distributed force fi (including the inertia force)which causes an additional stress s and an additional displacement ui from the initial state.The principle of virtual work for this initial stress problem is then derived as

gL

0

A(s0 + s)do ds−gL

0

(f0i + fi )dui ds=0, (6)

where L is the initial cable length and the summation convention is applied with respectto the subscript i. In equation (6), o is the axial strain from the initial state and is definedas [10]

o=dxi

ds1ui

1s+ 1

2

1ui

1s1ui

1s. (7)

Substitution of equations (5) and (7) transforms equation (6) into

gL

0

Asdo ds+gL

0

As01ui

1sdui

1sds−g

L

0

fidui ds=0. (8)

The first term of equation (8) gives the following expression for the strain energy in thecable due to the dynamic stress from the initial state:

Ua = 12 g

L

0

EAo2 ds. (9)

Here E is the Young’s modulus of the cable. The second term in equation (8), on the otherhand, would give the following potential energy (Ut ) due to the initial stress s0:

Ut = 12 g

L

0

As01ui

1s1ui

1sds. (10)

Note that only the non-linear part of the axial strain given by the second part of equation(7) constitutes the potential energy of the initial stress of the cable [5].

The bending strain energy of a cable is also evaluated in this study, in order to take intoaccount the energy dissipation due to bending deformation. The bending moment of acurved bar is related to the curvature change from the initial state, which is the same asan initially straight bar, and the bending strain energy can be represented by [11]

Ub = 12 g

L

0

EIk2 ds, (11)

where I is the cross-sectional moment of inertia and k is the curvature change of the cablefrom the initial state and is defined as

k= k*− k0. (12)

In equation (12), k* and k0 are the curvature of the vibrating cable configuration and thatof the initial cable configuration, respectively, and are obtained from [12]

k*2 =12(xi + ui )

1s2

12(xi + ui )1s2 , k2

0 =12xi

1s2

12xi

1s2 . (13a, b)

3.2.

In order to evaluate the modal potential energy in cables under flexural vibration, finiteelement analysis was conducted. A three-node non-linear cable element with a quadraticpolynomial shape function [13–15] was used. The static configuration of the cable was firstanalyzed non-linearly under the action of its self-weight, and then the eigenvalue problem

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was solved for small oscillations about the static equilibrium position. From the calculatednormal mode vectors, the additional axial strain, the curvature change, and hence eachcomponent of modal potential energy were calculated for each finite element.

The ith component of the initial static position vector xi at any point on the cableelement is expressed in terms of the nodal variable Xa

i at the node a as

xei (se)= s

3

a=1

Na(se)Xai , i=1, 2, 3, (14)

where the superscript e is used for element variables and Na (a=1, 2, 3) are the quadraticshape functions given in references [13–15]. The ith component of the nth modaldisplacement uni is then described in the same form with the calculated normal mode vector{fa

ni} and the natural frequency vn ,

ueni (se, t)= s

3

a=1

Na(se)Arfani cos vnt, i=1, 2, 3, (15)

where Ar is the reference amplitude of vibration. When the modal potential energy percycle is considered, only the time invariant part in equation (15) is to be used for theevaluation of the three components of the modal potential energy given in equations (9),(10) and (11) along with equations (7), (12) and (13). In this case, the dynamic strainamplitude should be carefully defined especially for highly sagged cables as was discussedin reference [7]. It is also worth mentioning that the integrations for each elementwere done by applying Gauss integration to obtain accurate values of the potentialenergy [16].

Once each component of the modal potential energy has been calculated for singlesuspended cables, the energy ratios which are necessary for the estimation of the modaldamping are easily calculated. The sag-to-span ratio is taken as one of the main parametersin the analysis of the modal energy ratios because modal characteristics of single suspendedcables are affected significantly by the sag-to-span ratio [9, 10]. Since the non-linearstrain–displacement relations are used for the evaluation of the axial strain and thecurvature associated with the modal vibrations, the amplitude of modal vibration is alsovaried in the analysis. A 200 m spanned steel cable with a diameter of 0·15 m was analyzedas an example, and, after checking the convergence of solutions, 40 elements were usedin the cable for the finite element analysis.

Figure 2. The modal energy ratio versus the sag ratio. (a) Axial strain energy ratio; (b) bending strain energyratio. ——, The first symmetric mode; – – – –, the first asymmetric mode; - - - - - -, the second symmetric mode;— – —, the second asymmetric mode.

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3.3.

The changes in the modal potential energy ratios with respect to the variation insag ratio were calculated and are shown in Figure 2 for the first four modes. Themodal amplitude of vibration was kept at 0·1% of the span length in this case. As isseen in Figure 2, the axial as well as the bending strain energy ratios, for asymmetricmodes, increase monotonically with increasing sag ratio. On the contrary, the energyratio of the symmetric modes has a relatively large increase in certain ranges of thesag ratio. These are known as the modal transition regions, where the lower symmetricmode of vibration changes into a higher one [9, 10]. It is also seen in Figure 2 that themagnitude of the axial energy ratio for the shallow cables, the sag ratio of which isless than 0·01, is at most of the order of 10−2 and that the bending energy ratio iseven smaller. This is because the main contribution to the total potential energy is fromthe initial tension, as depicted in Figure 3, where the comparison of each energycomponent, non-dimensionalized by EAL, is made for different modes. In most cases, thepotential energy of initial tension is several orders greater than the axial strain energy. Theonly exception is the axial strain energy of symmetric modes, in the modal transition regionof the sag ratio. The maximum value of the modal axial strain energy in the modaltransition region is almost the same in magnitude as the initial tension potential. It is alsoconfirmed, from Figure 3, that the axial strain energy is always larger than the bendingstrain energy, but the difference is not very large except for the symmetric modes in themodal transition region. Furthermore, the total potential energy of each mode decreasesmonotonically with increasing sag ratio, mainly because of a decrease in the magnitudeof the initial tension. This induces the increasing tendency of the energy ratios with sagratio in Figure 2.

Figure 3. The modal energy versus the sag ratio. (a) The first symmetric mode; (b) the first asymmetric mode;(c) the second symmetric mode; (d) the second asymmetric mode. ——, Total energy; - - - - -, potential energyof initial tension; – – – –, axial strain energy; — – —, bending strain energy.

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Figure 4. The modal energy ratio versus the modal amplitude of vibration. (a) Axial strain energy ratio;(b) bending strain energy ratio. ——, The first symmetric mode; – – – –, the first asymmetric mode; - - - - - -, thesecond symmetric mode; — – —, the second asymmetric mode.

The influence of the amplitude of vibration on the modal potential energy ratio is shownin Figure 4 for a sag ratio of 0·005. The abscissa of the figure is the reference amplitudeof vibration Ar , non-dimensionalized by the cable span length L*. In Figure 4(a) is showna non-linear dependency of the axial energy ratio on the amplitude of vibration. It is seenin Figure 4(a) that the axial energy ratio for the first mode is nearly constant, but for thehigher modes it increases with increasing the amplitude. This is because of the involvementof the nonlinear term in equation (7) for the axial strain–displacement relation. As is

Figure 5. The modal dynamic strain amplitude versus the modal amplitude of vibration. (a) The first symmetricmode; (b) the first asymmetric mode; (c) the second symmetric mode; (d) the second asymmetric mode.——, Linear; – – – –, non-linear.

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indicated in Figure 5, where comparisons between linear and non-linear components ofmodal dynamic strain for different modes are made, the non-linear strain amplitude of thefirst symmetric mode is always smaller than its linear value, whereas the non-lineardynamic strain amplitude of the first and the second asymmetric modes are always greaterthan their linear values. The linear values of dynamic strain amplitudes for asymmetricmodes are practically zero. The non-linear dynamic strain amplitude for the secondsymmetric mode of vibration, in Figure 5(c), is less than its linear value up to a certainamplitude of vibration and then it becomes much larger than the linear value for the higheramplitudes of vibrations. Therefore, it can be concluded that the linear strain is dominantonly in the case of the first symmetric mode, and the non-linear term of the axial strainexpression is the main source of dynamic strain for the second symmetric mode with largeramplitude and for the asymmetric modes.

As for the bending energy, it is evident from Figure 4(b) that the dependency of thebending strain energy ratio on the amplitude of vibration is very weak. This is becausethe dynamic curvature of the cable has a very weak nonlinear dependency on the amplitudeof vibration, and the vibration amplitude considered here is still small; hence thenon-linearity cannot be recognized even though the non-linear relation for the curvatureis used.

4. MODAL DAMPING OF STRUCTURAL CABLES

4.1.

The modal damping of ordinary or damping-untreated structural cable can now beinvestigated quantitatively by neglecting the bending term in equation (4). The per cycleenergy dissipation in ordinary structural cables has been experimentally investigated [17]and it is easily estimated from that study that the loss factor of structural cable has a valuebetween 0·04 to 0·06. Furthermore, it is seen in Figure 2 that the first mode axial energyratio, for the sag ratio of 0·005, which is a typical value for the cable in cable-stayedbridges, is about 8×10−2. Therefore, the modal damping ratio of the first symmetricmode, which gives the maximum modal damping, is estimated to be 0·002 (0·0125 inlogarithmic decrement) by taking the average value of the loss factor, namely 0·05. Sinceit is generally observed, from field experiments, that the modal damping of structuralcables in long cable-stayed bridges is about 0·01 or less, based on the logarithmicdecrement, the present analytical estimate of modal damping is very consistent. If therehad been no initial stress in the cable, the energy ratio would have been 1·0, which givesa modal damping ratio of 0·025, more than ten times greater than the damping ratio ofthe initially stressed cable. As was discussed earlier in this paper, the potential energy ofthe initial stress of the cable is very high and does not contribute to the energy dissipationin flexural vibration. Therefore, it is concluded that the modal damping of structural cablescannot be large because of the existence of the initial stress in the structural cables.

As can be seen from the energy-based definition of the modal damping in equation (4),there could be two alternative ways to increase the modal damping of strutural cables:either increase the modal energy ratios or increase the loss factors. The modal energy ratioscan be increased by increasing the sag ratio, because an increase in sag ratio decreases theamount of initial tension potential energy and hence gives higher energy ratio, as depictedin Figures 2 and 3. This in turn induces increased damping. However, increasing the sagratio is not recommended because it results in the reduction of the stiffness of the structure.Increasing the loss factors should, therefore, be considered in order to realize a highdamping cable, and the damping treatment shown in Figure 1 will be considered next.

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4.2. In the authors’ previous paper [1], the loss factors of damping treated structural cable

were discussed and explicit formulae for the axial loss factor and the bending loss factorwere derived analytically, by using the complex stiffness approach. The results aresummarized below.

The axial loss factor ha is obtained as

ha

hv=

ev

EA/(EA)cable ${4tv (tv +1)+1} 6 (1+2N)2

1+3N+3N27−1% , (16)

where hv is the loss factor of the viscoelastic material in the damping treatment, ev

(=Ev /Ecable ) is the ratio of the Young’s modulus of the viscoelastic material to that of thecable strand, tv (=Tv /d) is the thickness of the viscoelastic layer non-dimensionalized bythe diameter of untreated original cable without the outer cover (see Figure 1), and N isthe number of strand layers in the cable. The axial stiffness ratio of the damping treatedcable to the original cable, appearing in the denominator of equation (16), is given by

EA(EA)cable

=EAsv

EAcable+ {4evtv (tv +1)+4ecotco (tco +2tv +1)}6 (1+2N)2

1+3N+3N27 , (17)

where eco (=Eco /Ecable ) and tco (=Tco /d) are the Young’s modulus and the thickness ratiosof the outer cover, respectively.

The expression for the bending loss factor hb is given ashb

hv=

1EI/(EI)cable

(a1 − a2 + a3), (18)

witha1 = ev{(1+2tv )4 −1}, (19a)

a2 =4evtv6(1+ tv )2 +t2v

3+ 2

3 tv (1+ tv )7(1+2tv + tco )1

(1+ g)2 + (ghv )2 , (19b)

a3 =8ecotco6(1+2tv + tco )2 +t2co

37(1+2tv + tco )g

(1+ g)2 + (ghv )2 , (19c)

g=(G/4p2Ecotcotv )l2, l=(2L/nd), (19d, e)where Gv is the shear modulus of the viscoelastic material, l is the non-dimensionalwavelength, L is the length of the cable and n is the mode number. The bending stiffnessratio appearing in the denominator of equation (18) is given by

EI/(EI)cable =1+ ev{(1+2tv )4 −1}+ eco{(1+2tv +2tco )4 − (1+2tv )4}

−8evt2v6(1+ tv )2 +

t2v

3+ 2

3 tv (1+ tv )7 (1+2tv + tco )/tv

1+ g

−16ecotcotv6(1+2tv + tco )2 +t2co

37 (1+2tv + tco )/tv

1+ g. (20)

The axial loss factor ratio, as given by equation (16), along with equation (17), is plottedin Figure 6(a) for eco =1·0 and tco =0·05. The Young’s modulus ratio ev and the thicknessratio tv of the viscoelastic material are taken as the main parameters in Figure 6(a).Equation (18), defining the bending loss factor ratio, is also plotted in Figure 6(b) againstthe Young’s modulus ratio of the viscoelastic material for different non-dimensional

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Figure 6. The loss factor versus the Young’s modulus ratio of viscoelastic material [1]. (a) Axial loss factorratio. – – – –, tV =0·00; - - - - - -, tV =0·05; ——, tV=0·10. (b) Bending loss factor ratio. ——, l=2·67×103;– – – –, l=1·33×103; - - - - -, l=8·89×103; — – —, l=6·67×103.

wavelengths which are estimated for the first four vibration modes of a 200 m span cablewith a diameter of 0·15 m. The other parameters of the damping treatment are eco =1·0,tco =0·05 and tv =0·05. In the analysis, these parameters of the damping treatment wereselected as being practically reasonable and the loss factor of the viscoelastic material hv

is assumed to be 1·0 in the subsequent discussions. As can be seen in Figure 6, the bendingloss factor is much larger than the axial loss factor if a soft viscoelastic material, with smallYoung’s modulus, is used for the damping treatment. However, for a stiff treatment, thereis no difference in magnitude between the axial loss factor and the bending loss factor.This is in contradiction with the damping treatment applied to plates or beams, where aviscoelastic material with a relatively higher Young’s modulus is recommended in orderto achieve a higher damping effect. In cases of beams or plates, the main source of dampingin such a damping treatment lies in the shear deformation of the viscoelastic material. Inthe case of cables, however, it is not very easy to induce shear deformation in a relativelystiffer viscoelastic material because of the long wavelengths of vibration in cablesundergoing flexural vibrations. Therefore, if a stiffer viscoelastic material is used indamping treatment of long structural cables, the main source of damping becomes the axialdeformation rather than the shear deformation.

4.3.

The effectiveness of damping treatment of structural cables is investigated by estimatingthe increase of the modal damping ratio of the cable caused by the damping treatment.For this, the energy ratios previously obtained in Figures 2 and 4 are combined with theloss factors calculated and depicted in Figure 6. The results of the analysis for the modaldamping increment are shown in Figures 7 and 8. Sag ratios of 0·005 or 0·01 only areconsidered, because our main concern is the structural cable used in long spannedcable-stayed bridges. Although slight changes in the mode shapes can be identified withthe increase in the sag ratio even up to a sag ratio of 0·01, it is assumed, for simplicity,that the change in the mode shape or the wavelength is negligible.

In Figure 7 is depicted the increment of the modal damping ratio, in the damping treatedcable with the sag ratio of 0·005, subjected to an amplitude of vibration equal to 0·1%of the span length. The Young’s modulus ratio of the viscoelastic material, ev , is taken asa variable in Figure 7. As is seen in equation (4), the modal damping is a function of boththe modal potential energy ratios and the loss factors, and hence the tendency of the modal

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Figure 7. The modal damping ratio versus the Young’s modulus ratio of viscoelastic material. ——, The firstsymmetric mode; – – – –, the first asymmetric mode; - - - - -, the second symmetric mode; — – —, the secondasymmetric mode.

damping curve is a reflection of both the characteristics of the loss factor and thecharacteristics of the energy ratio. From Figure 7, it is seen that the modal damping, i.e.,the effectiveness of the damping treatment, is strongly dependent on the mode shape. Thisis not only because the bending loss factor is strongly dependent on the wavelength, butalso because the modal energy ratios are different for different modes. If vibration controlof the first mode is of most concern, then a stiffer viscoelastic material would be preferred,whereas a softer viscoelastic materal would be more effective in suppression of highermodes of vibration. From Figure 7, one sees that the maximum value of the modaldamping ratio for the case considered here is only about 0·0004, which corresponds to alogarithmic decrement of 0·003. It is generally observed, from wind tunnel experiments,that an additional damping required for suppression of rain–wind induced vibration ofcables, in cable-stayed bridges, is of the order of 0·01 or 0·02 (based on the logarithmicdecrement), which is equivalent to modal damping ratios of 1·6×10−3 or 3·2×10−3

respectively. Therefore, it should be concluded that the damping treatment proposed inreference [1] is not very effective in this case.

Figure 8. The modal damping ratio versus the modal amplitude of vibration. (a) Sag ratio=0·005; (b) sagratio=0·01. ——, The first symmetric mode; – – – –, the first asymmetric mode; - - - - -, the second symmetricmode; — – —, the second asymmetric mode.

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The effect of the amplitude of vibration on the modal damping of the damping treatedcable is shown in Figure 8, for ev =0·025. Figure 8(a) is for a sag ratio of 0·005 andFigure 8(b) is for a sag ratio of 0·01. It is clear that the modal damping curve reflects boththe characteristics of the loss factor and the characteristics of the energy ratio, and sincethe energy ratio has amplitude dependency, the modal damping is also dependent on theamplitude of vibration. The modal damping of higher modes increases and approaches thatof the first mode as the amplitude of vibration increases. Nevertheless, the effectivenessis very low for the cable with a sag ratio of 0·005 (Figure 8(a)). If the sag ratio of the cablecan be set as large as in Figure 8(b), then the additional modal damping can be increasedto 0·002 in damping ratio (or 0·013 in logarithmic decrement) and the damping treatmentin this case may have some value.

5. CONCLUSIONS

From the present study of the general damping characteristics of structural cables andthe effectiveness of damping treatments on the structural cables, the following conclusionscan be drawn.

(1) The modal damping ratio of a structural cable is defined in terms of the modalpotential energy ratios and the loss factors for the axial and bending deformations.

(2) The potential energy of the initial tension in shallow structural cables is much largerthan the axial strain energy, which in turn is larger than the bending strain energy.Therefore, the modal strain energy ratios in the energy-based damping definition generallyhave very small values, whereas the energy ratios are dependent on cable sag and theamplitude of vibration.

(3) Due to the ‘‘passive’’ behavior of the initial tension potential energy with respectto energy dissipation, and because of its very high magnitude, the damping of structuralcable is generally very low.

(4) The modal damping of the damping treated cable is strongly dependent on the modeshape. Stiff viscoelastic material is recommended for the damping treatment on thestructural cables if control of first symmetric mode is of main concern, whereas softviscoelastic material is more effective for higher mode control.

(5) The damping treatment formed by inserting a viscoelastic material into thestructural cable, which was proposed in the previous paper [1], does not give very highadditional damping to structural cables.

Despite conclusion (5), the damping treatment may have more value if either the energyratio or the loss factor of the damping treated structural cable is increased. For example,the modal damping of the damping treated cable can be increased by using viscoelasticmaterials with loss factors higher than the present assumed value of 1·0. Anotheralternative would be to apply layers of damping tapes spirally wound all around the wirestrands so that even in axial deformation of the cable, there is large energy loss due tothe shear deformation in the damping tape. This may be expected to yield an overall higheraxial loss factor of the damping treated structural cable, thereby increasing the modaldamping.

ACKNOWLEDGMENT

The authors would like to express their deep appreciation to Mr S. Shimada of SaitamaUniversity for carrying out the necessary calculations in this study.

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