5.12 Modal analysis

given $|\ddot{x}(t)⟩+M|x(t)⟩=0$, find the eigenvectors and eigenvalues of $M$. Then $\mathrm{\Phi }=[V_2,V_2]$ is $2\times 2$ matrix, transformation matrix. where each column is the eigenvector of $M$. Then $|X(t)⟩={\mathrm{\Phi }}^T|x(t)⟩$ and $|x(t)⟩=\mathrm{\Phi }$ $|X(t)⟩$. The new system becomes  $|\ddot{X}\left(t\right)⟩+\mathrm{\Omega }|X(t)⟩=0$ where $\mathrm{\Omega }$ is now diagonal matrix with eigenvalues of $M$ on the diagonal. Solve using this. First transform initial conditions to $X(t)$. Then trandform solution back to $|x(t)⟩$ using $|x(t)⟩=\mathrm{\Phi }$ $|X(t)⟩$.