From Computational View Point

The equation of motion for a damped SDoF (Single Degree of Freedom) system under given force is:

1.gif (1)

It can be shown that when we have no external force and system is subjected to ground excitation 11.gif, then the equation of motion turns to this form:

2.gif (2)


Let’s say we have n SDoF systems with mass of 12.gif, damping coefficient of 13.gif and stiffness of 14.gif for i’th system. Equation of motion for each system when subjected to ground excitation 11.gif is:

3.gif (3)

This differential equation can numerically solve by methods like Newmark-Beta or central difference or … then 31.gif will be obtained. Same approach can be use to obtain the 32.gif (ground speed) and 33.gif (ground displacement) from 11.gif. After solving the eq. 3 for all systems, we will have 35.gif, 36.gif and 37.gif and also 38.gif which is period of i’th system. Defining these parameters

4.gif (4)
5.gif (5)
6.gif (6)

The spectrum for i’th system is obtained by

7.gif
8.gif
9.gif

Then we will have 91.gif, 92.gif, 93.gif and 38.gif for every system and now can draw them into three charts of 94.gif against 97.gif, 95.gif against 97.gif and 96.gif against 97.gif,. Here is the example charts for this record:
* “The Imperial Valley (USA) earthquake of October 15, 1979.
* Source: PEER Strong Motion Database
* Recording station: USGS STATION 5115”

The top most chart is 94.gif which vertical axis is denoted by 98.gif, second one is 95.gif and third one is 96.gif.

chart.png

Last edited Aug 29, 2014 at 5:52 PM by epsi1on, version 4