## Monday, February 1, 2010

### Earthquake Model

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

(1) ,

where the forces acting on the i-th floor are

.

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) ,
where
.

A horizontal earthquake oscillation of amplitude of the form will produce an acceleration , and the opposite internal force on each floor of the building is . The resulting non-homogeneous system is

, where .

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 , and this system reduces to the form

Compute the eigenvalues of matrix ,
and the natural frequencies 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 .
Use the earthquake amplitude e = 0.075 ft = 0.9 in. for this example.

Solve the linear system using the parameters and e = 0.075.

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