Response Spectra

Although earthquake ground motion is complex, engineers require a simple characterization for use in design.
For this purpose they introduce the concept of response spectra.

Given a ground moiton acceereration histiry, x"(t), the response of a single degree of freedom damped oscillator, z(t), is governed by the seismograph equation

             z" + 2 ζωnz' + ωn2 z = - x"

where  ζ is the damping and ωn is the angular frequency of the oscillator which is related to the natural period period by the relation ωn =2π/Tn.
The concept of the response spectrum is to note the maximum displacement of the oscillator mass, |zmax | , or the maximum velocity, | z'max| , or the maximum acceleration, | z" max |.

A number of parameters can be defined:

Term
 Definition
Sd
 |zmax |
Sv
 |z'max|
Sa
 |z" max |
PSV (pseudo velocity)
 ωn Sd
PSA (pseudo acceleration )
 ωn 2 Sd



Example


Consider an impulse in acceleration passed through a oscillator with ζ=0.05 and natural periods  Tn = 0.1, 0.2, 0.3, 1.0, 2.0, 3.0, 10.0 and 20 seconds.
Figure 1 shows the resulting oscillator displacement z(t):

oscillator response
Fig. 1.  Oscillator response to an impulse in input acceleration.  The file name notation is Tn.Zeta.sac, e.g., 00.1.0.05.sac has a filter period of 0.1 sec and a damping of 0.05. 
 
The Fourier spectra of the displacement is

Spectra
Fig. 2.  Fourier spectra of the oscillator displacement. Note that at low frequencies, the displacement is ωn -2

Script:

these two figures were generated using the script  DOIT. This script creates a unit area impulse to represent the acceleration input.
For a given value of the damping and natural period, the input acceleration is input to the differential equation and the oscillator displacement is computed using Fourier transform techniques. NExt the script plots the traces and the Fourier amplitude spectrum of the trace.  Finally the trace determines the Sd for the trace.