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Single degree of freedom systems exist in various forms. A simple model is something like a lightpole, a slender pole with a heavy mass on top. Chopra's Example 6.2 is something similar to this structure. This page was last updated on Sept.6, 1999.
Learning Objectives: After completing this tutorial, the student can calculate the values of the linear elastic displacement, psuedo-velocity, and psuedo-acceleration response spectra for a given time-history ground motion. Note that the procedure uses a computer to determine the displacement time-history of the system, but if the ground motion was simple to define mathematically (perhaps as a sine function), the entire process could be done by hand!
Developing a Response Spectra using the lightpole model Chopra (Section 6.6.5) shows the steps required to develop a Response Spectra.
Step 1: Define the ground acceleration (we will
use the El Centro, 1940 ground motion).
Step 2: Select the natural period vibration and damping ratio
Step 3: Compute the displacement time history for this model. We use SAP2000 to run the analysis, although any linear dynamic analysis will work.
Step 4: Determine the peak displacement of this time history
Step 5: Determine the spectral ordinates for displacement,
psuedo-velocity, and psuedo-acceleration.
Step 6: Repeat Steps 2 through 5 for different sdof systems
(vary the height of the lightpole) First, use the 8'-8" tall lightpole = period of 1.0 seconds
Next, use the 5'-7" tall lightpole = period of 0.5 seconds We can continue this process, until we collect data for as many different periods as we want, and we can also choose different amounts of damping.
Step 7: Present results graphically. Finally we have calculated the values for all possible values of the
period and obtained a complete graph of the displacement, psuedo-velocity, and
psuedo-acceleration response spectra. Click here for a picture of the complete
Displacement Response Spectra
Now the process could be repeated for different values of damping. The same procedure can be used to develop a response spectrum for any source of vibration. For example, wind storms, blast loading, impact or a harmonic vibration due to a motor.
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