Monday, February 1, 2010

Earthquake Model

In the study of earthquake induced vibrations on multistory buildings, the free transverse oscillations satisfy the equation

(1) [Graphics:Images/EarthQuakeModelMod_gr_1.gif],

where the forces acting on the i-th floor are

[Graphics:Images/EarthQuakeModelMod_gr_2.gif].

Consider a building with n floors each of mass m slugs and the horizontal restoring force of k tons/foot between floors. Then this system reduces to the form

(2) [Graphics:Images/EarthQuakeModelMod_gr_3.gif],
where
[Graphics:Images/EarthQuakeModelMod_gr_4.gif] .

A horizontal earthquake oscillation of amplitude [Graphics:Images/EarthQuakeModelMod_gr_39.gif] of the form [Graphics:Images/EarthQuakeModelMod_gr_40.gif] will produce an acceleration [Graphics:Images/EarthQuakeModelMod_gr_41.gif], and the opposite internal force on each floor of the building is [Graphics:Images/EarthQuakeModelMod_gr_42.gif]. The resulting non-homogeneous system is

[Graphics:Images/EarthQuakeModelMod_gr_43.gif], where [Graphics:Images/EarthQuakeModelMod_gr_44.gif].

Example 1. Consider a building with n = 6 floors each of mass m = 1250 slugs (weight of 20 tons)
and the horizontal restoring force of k = 10,000 lb/ft = 5 tons/foot between floors.
Then [Graphics:Images/EarthQuakeModelMod_gr_5.gif], and this system reduces to the form

Compute the eigenvalues of matrix [Graphics:Images/EarthQuakeModelMod_gr_6.gif],
and the natural frequencies [Graphics:Images/EarthQuakeModelMod_gr_7.gif] and periods P of oscillation of the building.

Example 2. Solving the above non-homogeneous system for the coefficient vector v for X[t].
The vector v is the solution to the equation [Graphics:Images/EarthQuakeModelMod_gr_45.gif].
Use the earthquake amplitude e = 0.075 ft = 0.9 in. for this example.

Solve the linear system using the parameters [Graphics:Images/EarthQuakeModelMod_gr_46.gif] and e = 0.075.

Find the coefficient vector v and the vector X[t]. Plot the vibrations of each floor.

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Introductory Methods of Numerical Analysis