Module for the study of the Earthquake Model of Induced Vibrations

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

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
          

Example 1.  Compute the eigenvalues of matrix  ,
and the natural frequencies    and  periods  P  of oscillation of the building.

Background.

    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 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.

Example 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    over the interval  ,  
this graph should have six vertical asymptotes corresponding to each value    in the table above.  
Then plot the maximum amplitude as a function of the period  p  in seconds.

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