Chapter 28, Problem 29P

Engineers and scientists use mass-spring models to gain insight into the dynamics of structures under the influence of disturbances such as earthquakes. Figure P28.29 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

$\left(\frac{k_1+k_2}{m_1}-w^2\right)X_1$ $-\frac{k_2}{m_1}{X}_2=0$

$-\frac{k_2}{m_2}$ $X_1+\left(\frac{k_2+k_3}{m_2}-w^2\right)X_2-\frac{k_3}{m_2}{X}_3=0$

$-\frac{k_3}{m_3}{X}_2+\left(\frac{k_3}{m_3}-w^2\right)X_3=0$

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

FIGURE P28.29