Engineers and scientists use mass-spring models to gain insight into the dynamics of structures under the influence

Question:

Engineers and scientists use mass-spring models to gain insight into the dynamics of structures under the influence of disturbances such as earthquakes. Figure shows such a representation for a three-story building. For this case, the analysis is limited to horizontal motion of the structure; Force balances can be developed for this system as

k2 k + k2 X2 k3 x, kz + k3 Xi + +(**-)*-X=0 k2 k3 X2 +

Determine the eigenvalues and eigenvectors and graphically represent sent the modes of vibration for the structure by displaying the amplitudes versus height for each of the eigenvector. Normalize the amplitudes so that the displacement of the third floor isone.