Earthquake Model

Background.

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 withn floors each of massmslugs and the horizontal restoring force ofk tons/footbetween floors. Then this system reduces to the form

(2),
where
.

Example 1.Consider a building withn = 6floors each of massm = 1250 slugs (weight of 20 tons)
and the horizontal restoring force ofk = 10,000 lb/ft = 5 tons/footbetween floors.
Then,and this system reduces to the form

Compute the eigenvalues of matrix,
and the natural frequenciesandperiodsPof oscillation of the building.

More Background.

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

,where.

Example 2.Solving the above non-homogeneous system for the coefficient vectorvforX[t].
The vectorvis the solution to the equation.
Use the earthquake amplitudee = 0.075 ft = 0.9 in.for this example.

Solve the linear system using the parameters ande = 0.075.

Find the coefficient vectorvand the vectorX[t].Plot the vibrations of each floor.

Example 3.For the above non-homogeneous system the coefficient vectorvis the solution to the equation

.

Plot the maximum amplitude of oscillation of the floors vs the parameterover the interval,
this graph should have six vertical asymptotes corresponding to each valuein the table above.
Then plot the maximum amplitude as a function of the periodpin seconds.