This paper proposed an evaluation formula of the reduction factor of the total input energy EFh due to the damping in the elastic single-mass system. Based on the theoretical formula at the stationary response, the fundamental formula for non-stationary response was defined and its coefficient was statistically determined by response analysis. As a result, the evaluation formula of EFh was induced as functions of the damping factor and period of the system.
The major findings obtained in this paper were follows.
1) The reduction factor of the total input energy EFh due to the damping in the elastic single-mass system is given in the function of β which means the ratio of the velocity energy corresponding to the damping h0 and h, as follows: EFh=Eh/E0=β·(h/h0). The β is the ratio of the integration of the square of response velocity corresponding to the damping h0 and h.
2) At stationary response, the ratio β is given by (h0/h)^2.
3) At non-stationary response, the ratio β is expressed as β=(h0/h)^(2·α). Here, the α is a function of the ratio of the period T/T0 (T is the period of the system, and T0 is the boundary period of the region which is so called as constant velocity).
4) In the case of the application of the approximate values, defined in notification, as the amplification factor Gs of the ground surface, the α was given as α=0.10·T/T0+0.51 when T/T0 was less than 1, and the α was given as α=0.21·T0/T+0.40 when T/T0 was 1 or more, regardless of the class of soil.
5) In the case of the application of the precise calculation, defined in notification, as the amplification factor Gs, the analytical response results depend on class of soil, so the correction factor γ according to the class of soil was induced into the formula.
6) The evaluation formula EFh was given as EFh=(h0/h)^(0.20·T/T0+0.02) when T/T0 was less than 1, and the EFh was given as EFh=(h0/h)^(0.42·γ· T0/T+0.22−0.42·γ) when T/T0 was 1 or more. The correction factor γ was approximately 1.0 in the case of approximate values of Gs, and was approximately 1.0, 1.59 and 2.37 for the class of soil 1, 2 and 3, respectively, in the case of precise calculation of Gs. The proposed formula of the reduction factor of the total input energy EFh showed an excellent relationship with the analytical response results.