同济大学：《结构动力学》课程教学资源（教案讲义）Lecture 04 Response to harmonic and periodic excitations - Application

Equivalent viscous damping Equivalent viscous damping:to equate the energy dissipated in a vibration cycle of the actual structure and an equivalent viscous system For an actual structure the force-displacement relationship obtained from an experiment under cyclic loading with displacement amplitude can be determined. 目 土本程李悦

 Equivalent viscous damping • Equivalent viscous damping: to equate the energy dissipated in a vibration cycle of the actual structure and an equivalent viscous system • For an actual structure the force-displacement relationship obtained from an experiment under cyclic loading with displacement amplitude can be determined

▣f-u Curve =ci(t)=cou cos(ot-p) =coyu2-[uosin(ot-p)=cou2-u(t) {61m a>0 ↑Colo The area of the ellipse equals Ep i<0 Sp=πab=π(4o)(cou） =元c0u,2=250ku,2=E。 目 土本2程季院

        0 2 2 2 2 0 0 0 cos sin D f cu t c u t c u u t c u u t                     fD-u Curve  The area of the ellipse equals ED 2 2 0 0 1 D u f u c u                ellipse  0 0   2 2 0 0 2 D D n S ab u c u c u ku E             

口 Total (elastic plus damping)resisting force- displacement curve fo+fs fs=ku fp i>0 ↑coo Loading,u>0 i<0 、Unloading,W<0 (a) b 闺 土本程李悦

 Total (elastic plus damping) resisting force – displacement curve

Equivalent viscous damping En=250ku2=45% =4π5 Eso On 5= 1 Ep 4元0nEs0 1 Ep 4元Eso o E Deformation 目 土本2程季院

 Equivalent viscous damping 2 2 0 0 0 0 1 2 4 4 2 1 4 D S n n n n D S E ku ku E E E                       

Non-viscous damping Hysteretic damping model(Rate-independent damping): The damping force is proportional to velocity and inversely proportional to frequency a器 Clough 1981 =inku(t)clough 1993 -Chopra 1995 闺 土本程李悦

      D u t f k u t u t  f i ku t D     D   k f u t    ——Clough 1993 ——Chopra 1995 ——Clough 1981  Non-viscous damping • Hysteretic damping model (Rate-independent damping): The damping force is proportional to velocity and inversely proportional to frequency