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 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
         .  

Proof  Earthquake Models  Earthquake Models  

Computer Programs  Earthquake Models  Earthquake Models  

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.
Solution 1.

More 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.
Solution 2.

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.
Solution 3.

Old Lab Project (Earthquake Induced Vibrations  Earthquake Induced Vibrations).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Earthquake Models  Earthquake Models  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Earthquake Induced Vibrations

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(c) John H. Mathews 2004