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The objectiveof this design is to generate an interaction between the inner BFRPtendons and the outer hybrid B/CFRP 50% tendons when the entirecable is excited to vibrate because of the different dynamic charac-teristic of these two portions of cables, as shown in Fig. 2. Thisinteraction can act on the inserted viscoelastic material, which willdissipate vibration energy intelligently with respect to the occur-rence and amplitude of vibration.2.2. Arrangement along the cableBased on above sectional design of smart dampers, the distribu-tion of inserted viscoelastic material is discussed according to itsminimum influence on the cable mechanical behavior and themaximum effect on dissipation of vibration energy.Because the inserted viscoelastic material can be regarded as anadditional weight applied along the cable due to its much lowerelastic modulus compared with the cable material, a large amountinserted will lead to a negative effect on mechanical behavior suchas the sag and k2(an integrated parameter for nonlinearity of cabledefined by Irvine [21]). For instance, for a 575 mlength of hybrid B/CFRP 25% cable with a 30 mm thickness of continuous viscoelasticmaterial, the sag at the middle span will increase by 42% and k2ofthe cable will increase by 100%, which will reduce the static anddynamic advantages of hybrid FRP cable in long-span bridges.Moreover, this influence will become even more critical withincreasing cable length. Thus, a discontinuous distribution is sug-gested as an alternative. By introducing a discontinuous distribu-tion, the amount of inserted viscoelastic material will be greatlyreduced and the corresponding influence to the cable’s static anddynamic behavior will be minimized. Foremost, the energy dissi-pated by the discontinuous distribution can also be equivalent to that dissipated by the continuous distribution, which is explainedby the following comparison.The energy dissipated by a viscoelasticmaterial per unit volumein one cycle can be represented by the following equation [22]:Q ¼Ir0de ¼Z T0r0dedtdt ¼ pxE00e20 ð1Þwhere E00is the storage modulus of the viscoelastic material. e0 andr0 are themaximumstrain and stress in one cycle, respectively. x isthe circular frequency of vibration.For the continuous distribution (Fig. 3a), the total energy dissi-pated by the viscoelastic material can be expressed byQ1 ¼ V1pxE001e20 ¼ AcLpxE001F1E01Lb 2ð2Þwhere V1 is the total volume of continuous viscoelastic material, Acis the cross-sectional area of viscoelastic material, and L is thelength of cable, F1 is the total interaction between inner and outercable, E0is the elastic modulus, b is the depth of viscoelastic mate-rial in transversal direction of cable.For discontinuous distribution (Fig. 3b), the total energy dissi-pated by viscoelastic material can be expressed byQ2 ¼ V2pxE002e20 ¼ AcNlpxE002F2E02Nlb 2ð3Þwhere V2 is the total volume of the discontinuous viscoelastic mate-rial, l is the length of an inpidual viscoelastic damping material, Nis the number of viscoelastic damping elements, and F2 is the totalinteraction between the inner and outer cable.Because the vibration energy is dissipated by compression, theallowable maximum strain of the viscoelastic material is limited.Since the total interaction force between the inner and outer cableis determined only by the properties of the inner and outer cableand the amplitude of vibration, F1 equals F2. Thus, lettingF1E01Lb¼ F2E02Nlb, and the following is obtained:E01L ¼ E02Nl ð4ÞAs Nl is much smaller than L, E02 should be much larger than E01 tomaintain the equality. Substituting Eq. (4) into Eqs. (2) and (3),the ratio of Q1 and Q2 is Therefore, it is indicated from Eq. (5) that the dissipated energy ofdiscontinuous dampers can be equivalent to that of continuousdampers by choosing the viscoelastic materials with larger elasticmodulus.3. Theoretical analysis of modal damping3.1. AssumptionsTo evaluate the damping effect of stay cable, it is generallymoreconvenient and physically reasonable to define the damping of amulti-degree of freedom (MDOF) system using damping ratio foreach mode rather than to evaluate the coefficients of the dampingmatrix because the modal damping ratios can be determinedexperimentally or estimated with adequate precision [23]. Basedon the previous studies of dynamic characteristic of hybrid B/CFRPcable for long-span cable-stayed bridge, only the first order of cablemode has a potential risk of resonance between the cable andbridge [17]. Thus, in this study, only the first order of vibration isadopted to evaluate the damping effect of this smart damper forhybrid B/CFRP cable.Given that the objective is to evaluate the damping effect of thesmart dampers, only the internal damping generated by the in-serted viscoelastic material will be considered; the other sourcesof damping including material damping of the cable, frictionamong each tendon and aerodynamic damping are not taken intoconsideration. The energy dissipated by axial and bending defor-mation of the viscoelastic material will also be neglected due toits discontinuous distribution along the cable.3.2. Energy based dynamic equilibrium equationsThe dynamic response of cable can be directly described by alogarithmic decrement of damping which can be derived by apply-ing Hamilton’s principle [23]:Z T2T1dðPT PV ÞdT ¼ Z T2T1dWdT ð6Þwhere PT, PV andWdenote the cable kinetic energy, cable potentialenergy and the work done by nonconservative force, respectively.The cables in cable-stayed bridge are applied with an initial tensionin order to maintain the static equilibrium. This initial tension givesrise to the geometrical stiffness that constitutes a dominant portionof the total stiffness of the cable. Therefore, the potential energy ofthe cable consists of strain energy generated by the initial tensionand vibration and gravitational potential energy. Herein, when weuse the free-vibration decay method to determine the modal damp-ing ratio, the work done by nonconservative forces is equivalent tothe energy dissipated by the viscoelastic material, as show below: Z T2T1dWdT ¼Z T2T1QðtÞdT ð7ÞThe potential energy