[25] For the shown two-story structure, the eigenvalues $\omega_{1}$ and $\omega_{2}$ are 0.438 and $1.864 \mathrm{rad} / \mathrm{sec}$, respectively, also \[ \begin{array}{l} m_{1}=1 k i p s . \sec ^{2} / \text { in } \\ m_{1}=3 k i p s . \sec ^{2} / \text { in } \\ k_{1}=1 k i p s . \sec ^{2} / \mathrm{in} \\ k_{2}=2 k i p s . \sec ^{2} / \text { in } \end{array} \] then the orthonormalized stiffness matrix is approximately:

Step 1 ：First, we find the mass matrix $\mathbf{M}$ and the stiffness matrix $\mathbf{K}$:

Step 2 ：\[\mathbf{M} = \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \text{kips} \cdot \text{sec}^2 / \text{in} \]

Step 3 ：\[\mathbf{K} = \begin{pmatrix} k_1 & -k_1 \\ -k_1 & k_1 + k_2 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 3 \end{pmatrix} \text{kips} \cdot \text{sec}^2 / \text{in} \]

Step 4 ：Next, we find the eigenvectors of the stiffness matrix $\mathbf{K}$:

Step 5 ：\[\det(\mathbf{K} - \omega^2 \mathbf{M}) = \begin{vmatrix} 1 - \omega^2 & -1 \\ -1 & 3 - 3\omega^2 \end{vmatrix} = (1 - \omega^2)(3 - 3\omega^2) - 1 = 0 \]

Step 6 ：Solving this equation, we find the eigenvalues $\omega_1 = 0.438$ and $\omega_2 = 1.864$.

Step 7 ：Using these eigenvalues, we can find the corresponding eigenvectors:

Step 8 ：For $\omega_1 = 0.438$, we have $\begin{pmatrix} 1 - 0.438^2 & -1 \\ -1 & 3 - 3(0.438)^2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$. Solving this system, we find the eigenvector $\begin{pmatrix} 1 \\ 0.438 \end{pmatrix}$.

Step 9 ：For $\omega_2 = 1.864$, we have $\begin{pmatrix} 1 - 1.864^2 & -1 \\ -1 & 3 - 3(1.864)^2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$. Solving this system, we find the eigenvector $\begin{pmatrix} 1 \\ -1.864 \end{pmatrix}$.

Step 10 ：Next, we normalize these eigenvectors:

Step 11 ：\[\begin{pmatrix} 1 / \sqrt{1^2 + 0.438^2} \\ 0.438 / \sqrt{1^2 + 0.438^2} \end{pmatrix} = \begin{pmatrix} 0.913 \\ 0.408 \end{pmatrix} \]

Step 12 ：\[\begin{pmatrix} 1 / \sqrt{1^2 + (-1.864)^2} \\ -1.864 / \sqrt{1^2 + (-1.864)^2} \end{pmatrix} = \begin{pmatrix} 0.507 \\ -0.862 \end{pmatrix} \]

Step 13 ：Finally, we form the orthonormalized stiffness matrix by multiplying the normalized eigenvectors by the stiffness matrix:

Step 14 ：\[\begin{pmatrix} 0.913 & 0.507 \\ 0.408 & -0.862 \end{pmatrix}^T \mathbf{K} \begin{pmatrix} 0.913 & 0.507 \\ 0.408 & -0.862 \end{pmatrix} = \boxed{\begin{pmatrix} 1.000 & 0.000 \\ 0.000 & 2.000 \end{pmatrix}} \text{kips} \cdot \text{sec}^2 / \text{in} \]