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PV can be represented by the summationof PS and PG:PS ¼ PiS þZ L0Ped þ 12EAe2d dS ð8ÞPG ¼ PiG þZ L0 mgUvdS ð9Þwhere PiS and PiG are the strain energy and the gravitational poten-tial energy under static equilibrium, respectively [24]. P is the staticcable force. E and A are the elastic modulus and cross-sectional areaof the cable, respectively. ed is the dynamic strain generated by vibration. The cable weight per unit length is given by mg. Uv is thedisplacement along the vertical direction. L is the cable length.PT can be expressed asPT ¼Z L012mV21 þ V22 þ V23 dS ð10Þwhere V1, V2 and V3 are the velocities of each element in three direc-tions, respectively.The energy dissipated by the viscoelastic material per unit vol-ume in one cycle is represented by Eq. (3).Considering the difference of the amplitudes within n cycles,the dynamic equilibrium equation in Eq. (6) can be expressed asZ nT0ðDPT DPV ÞdT ¼ Z nT0dWdT ¼Z nT0QðxÞdT ð11ÞThe initial status is set to be the firstmaximumamplitude of thevibration after the release of external force/displacement, and thefinal status is chosen to be the maximum amplitude of the vibra-tion after n cycles (periods). Because the kinetic energy is main-tained at zero for both statuses, the only variation of thepotential energy is induced by the strain energy and gravitationalpotential energy, which are only related to the initial and final sta-tuses as they belong to the conservative forces. Thus, the variationof the cable potential energy can be represented byDPS ¼Z L0Pðed;1 ed;nÞþ 12EA e2d;1 e2d;n dS ð12ÞDPG ¼Z L0 mgðUv;1 Uv;nÞdS ð13ÞDPS þ DPG ¼X n1Z L0QðSÞdS ð14Þwhere ed,1 and ed,n represent the dynamic strain at the first cycleand at n cycles, respectively. Uv,1 and Uv,n represent the vertical dis-placement at the first cycle and at n cycles, respectively.Considering the different characteristics of in-plane and out-of-plane vibrations of the cable, the corresponding logarithmic decre-ment of damping can be derived, respectively.3.3. Derivation of modal damping ratio3.3.1. In-plane vibrationHerein, we consider a stay cable with an inclination of h with re-spect to the horizon and assume a constant dynamic strain alongthe cable. The assumption is usually adopted for the analysis ofcable vibration [25,26]. For a small sagging cable (d/L < 1/8), thestatic configuration of the cable can be described by the parabolafunction with sufficient accuracy:y ¼ 4dxL1 xL ð15Þwhere d is the sag at mid-span perpendicular to the longitudinaldirection of the cable, as shown in Fig. 4, given by d = mgL2cos h/8P [27].Due to the sag effect on in-plane vibration, the first order ofvibration is represented by [11]UiðxÞ¼ Um;ið1 tanðx0=2Þ sinðx0x=LÞ cosðx0x=LÞÞ ð16Þwhere Um,i is the maximum amplitude at the middle span withinthe i cycle, x is given by x0= xL/(P/m)1/2.Thus, the variation ofthe gravitational potential energy within n cycles can be integratedalong the cable direction and represented as DPG ¼Z L0 mgðU0ðxÞ UnðxÞÞdx ¼ 2pmgL cos hðUm;0 Um;nÞð17ÞIn terms of in-plane equilibrium, the dynamic stain can be ex-pressed as [2]ed;i ¼ mg cos hPLeZ L0UiðxÞdx ð18Þwhere Le = L[1 + 8(d/L)2].Substituting Eq. (16) into Eq. (18), the dynamic strain is ex-pressed byed;i ¼ gUm;i ð19Þwhere g ¼ mg cos hPx0ð1þ8ðd=LÞ2Þx0þ tan x02 cos x02 x sin x02 x .Thus, the variation of strain energy within n cycles is obtainedby substituting Eq. (19) into Eq. (12)DPS ¼ PgðUm;0 Um;nÞþ 12EAg2ðU2m;0 U2m;nÞ L ð20ÞThe energy dissipated by the viscoelastic material, as shown inthe right side of Eq. (14), can be obtained from Eq. (3) and ex-pressed asQ ¼X n1AcNlpxE00FiE0Nlb 2ð21ÞFi is the maximum interaction between the inner and outer cable inthe ith cycle. Newton’s equation of motion is adopted to representthe interaction between the inner and outer cable with respect totime and position.Considering the dynamic equilibrium of a segment of innercable in the y direction, as shown in Fig. 5, the equilibrium equa-tion can be represented [25]:P@2u@x2þ hs;i@2y@x2¼ mcos h@2u@t2 fi ð22Þwhere hs is the dynamic portion of cable force obtained by ed,iEAand fi is the uniform interaction between the inner and outer cable.The transversal motion with respect to time and the x coordi-nate can be described by the following product [26]:uðx; tÞ¼ UiðxÞeixtð23ÞSubstituting Eq. (23) into Eq. (22):fi ¼ ½aeixtþ b ð24Þwherea¼mUm;ix2ðcoshþðtanðx0=2Þsinðx0x=LÞþcosðx0x=LÞÞð1 coshÞÞ