I'm reading a book of Anil. K. CHOPRA related to Structural Dynamics, I'm intending to obtain the deformation response spectrum from given ground motion. What I have here in hand is ug(t) ground motion, but I couldn't derive the deformation response from that. In book section 6.4 Response History mentions that" For a given ground motion ug(t), the deformation response u(t) of and SDF system depends only on the natural vibration period of the system and its damping ratio"
In the very same book there are sections which describes the Direct Integration methods such as "Methods Based On Interpolation of Excitation" to find the deformation response but in the formulation it includes the k spring coefficient of SDF system, that contradicts with previous statement in the book that deformation response depends on natural period and damping ratio.
Methods Based On Interpolation of Excitation
Here is the method that is mentioned by Author in the book, it is time integration method and since it has an example in it, it is very convenient for me to understand, everything is conceivable the glitch here is "C" and "D" coefficients includes "k" in their formula. I couldn't strip down that "k" because I don't know it, I was expecting simply 𝜉 and $T_n$.
So at the end how am I suppsed to obtain the deformation response history from given seismic ground motion?