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  [Graphics:Images/EarthQuakeModelMod_gr_5.gif],  and this system reduces to the form  

Compute the eigenvalues of matrix  [Graphics:Images/EarthQuakeModelMod_gr_6.gif],
and the natural frequencies  [Graphics:Images/EarthQuakeModelMod_gr_7.gif]  and  periods  P  of oscillation of the building.

Solution 1.

Enter the matrix  a  into Mathematica.

[Graphics:../Images/EarthQuakeModelMod_gr_8.gif]



[Graphics:../Images/EarthQuakeModelMod_gr_9.gif]


Compute the eigenvalues [Graphics:../Images/EarthQuakeModelMod_gr_10.gif] of the matrix  a, and the natural frequencies  [Graphics:../Images/EarthQuakeModelMod_gr_11.gif]  and  periods  p  of oscillation of the building.

[Graphics:../Images/EarthQuakeModelMod_gr_12.gif]


   λ

[Graphics:../Images/EarthQuakeModelMod_gr_13.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_14.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_15.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_16.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_17.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_18.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_19.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_20.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_21.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_22.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_23.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_24.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_25.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_26.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_27.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_28.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_29.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_30.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_31.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_32.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_33.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_34.gif]

[Graphics:../Images/EarthQuakeModelMod_gr_35.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_36.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_37.gif]

  

[Graphics:../Images/EarthQuakeModelMod_gr_38.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004