Background. In the study of earthquake induced vibrations on multistory buildings, the free transverse oscillations satisfy the equation , where the forces acting on the k-th floor are . 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 , where

Computer Lab Work.

Exercise 1. Compute the eigenvalues of matrix a, and the natural frequencies w and periods P of oscillation of the building.

More Background. A horizontal earthquake oscillation of amplitude e 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 .

Exercise 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 Exercise.

Solve the linear system using the parameters w = 2.1 and e = 0.075. Find the coefficient vector v and the vector X[t]. Plot the vibrations of each floor.

First, enter the column vector b.

Second, create the matrix .

Print the linear system we want to solve.

Third, solve the linear system using w = 2.1 and e = 0.075.

Fourth, find the maximum amplitude of oscillation of the floors in feet and in inches
On what floor did the maximum amplitude of oscillation occur ?

Fifth, find the minimum amplitude of oscillation of the floors in feet and in inches
On what floor would did the minimum amplitude of of oscillation occur ?

Sixth, form X[t], but plot X[t] + k for floor k. Do this on one graph by forming a set of functions to plot parametrically.

Now plot the functions.
Imagine a vertical line through x = 1,2,3,4,5,6 which would represent no movements of the floors.

Exercise 3. For the above non-homogeneous system the coefficient vector v is the solution to the equation .

Plot the maximum amplitude of oscillation of the floors vs the parameter w over the interval 0 <= w <= 6, this graph should have six vertical asymptotes corresponding to each value w in the table above. Then plot the maximum amplitude as a function of the period P in seconds.

First, create the following function of the two variables w and e. Do this carefully, each time Mathematica plots one point on the graph it will need to solve a 6 by 6 linear system and find the maximum of the absolute value of the entries in the vector v.

Test your function with the information you had in Exercise 2.
g[2.1, 0.075]

0.388417

Now plot g[w, 0.075] and observe that there is a vertical asymptote for each of the values w in the table in Exercise 1.

Now make two plots for g[w, 0.075] as a function of the period P in seconds.

What is a "bad time" for the period of an earthquake ?

Solutions.

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(c) John H. Mathews, 1998