The typical earthquake ground motion response spectrum represents an envelope of the peak responses of many single-degree-of-freedom (SDOF) systems with different periods. The acceleration response spectrum of a ground motion is a relationship between the natural period of vibration of a SDOF system and the maximum absolute acceleration that it experiences under the ground motion. Similarly, a displacement response spectrum typically represents the peak displacement, relative to the ground, of many SDOF systems with different periods. Hence, the construction of a response spectrum involves the analysis of many different SDOF systems. The value of each point on the spectrum is the peak response of a single degree of freedom system of a given period. This is illustrated in Figure below, where the displacement spectrum of a record from the Palm Springs Earthquake is shown, along with the time history response at several periods. The relationship between the peak response at the different periods and the spectrum is graphically illustrated. 

Figure: Construction of linear displacement spectra for a record from the 1986 North Palm Springs earthquake. The displacement histories are shifted to make the peak displacements coincide with the time origin.

The most commonly used forms of linear spectra are:

•Acceleration spectra “Sa” (Sa vs. T),

•Displacement spectra “Sd” (Sd vs. T),

•Velocity spectra “Sv” (Sv vs. T).

Other commonly used spectra are pseudo-acceleration spectra “PSa” and pseudo-velocity spectra “PSv”, which are easily derived from the displacement spectra using the following equations:

PSv = ω * Sd

PSa = ω * PSv  = ω2 * Sd

Figure below shows plots of PSa, PSv, and Sd vs. T. Also shown is a plot of PSa vs. Sd, which is another spectral representation commonly referred to as the ADRS spectrum.

Figure: PSa, PSv, Sd and ADRS spectra for the 1994 Northridge Rinaldi record, for multiple damping values.

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