Structures with additional frictional damping system have strong nonlinearity that the dynamic behavior is highly affected by the relative magnitude between frictional force and excitation load. In this study, normalized response spectra of the structures with non-dimensional friction force are obtained through nonlinear time history analyses of the mass-normalized single degree of freedom systems using 20 ground motion data recorded on rock site. The variation of the control performance of frictional damping system is investigated in terms of the dynamic load and the structural natural period, of which effects were not considered in the previous studies. Least square curve fitting equations are presented for describing those normalized response spectrum and optimal non-dimensional friction forces are obtained for controlling the peak displacement and absolute acceleration of the structure based on the derivative of the curve-fitted design spectrum.